CalcPlot3D

Discuss interdimensional programming, Java applets and so forth.

CalcPlot3D

Postby ICN5D » Tue Apr 08, 2014 3:49 am

This is the awesome program I use to make the toratope renders and animations. I have made several explorer scripts for the 3D torus and all 4D toratopes, so far. I'll be uploading them here if anyone wants to use them. Attached is a Step description of each one through 4D.


http://web.monroecc.edu/calcNSF/


3,4D Step Descriptions.txt
(805 Bytes) Downloaded 86 times

3D Torus ((II)I).txt
(6.1 KiB) Downloaded 69 times

4D Glome, 4-ball (IIII).txt
(3.35 KiB) Downloaded 65 times

4D Torisphere ((III)I).txt
(5.04 KiB) Downloaded 65 times

4D Spheritorus ((II)II).txt
(5.06 KiB) Downloaded 61 times

4D Ditorus (((II)I)I).txt
(10.13 KiB) Downloaded 60 times

4D Tiger ((II)(II)).txt
(5.1 KiB) Downloaded 69 times
Last edited by ICN5D on Wed Jul 23, 2014 11:00 pm, edited 1 time in total.
in search of combinatorial objects of finite extent
ICN5D
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Posts: 1044
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Re: CalcPlot3D

Postby ICN5D » Mon Jun 02, 2014 5:55 am

Here's a bunch more exploration scripts for some wild, far-out, high-D shapes:

5D Tritorus ((((II)I)I)I).txt
(7 KiB) Downloaded 63 times

6D Double Tiger (((II)(II))(II)).txt
(3.58 KiB) Downloaded 58 times

6D Spheritigritorus ((((II)I)(II)I).txt
(3.48 KiB) Downloaded 58 times

6D Tigriditorus ((((II)I)I)(II)).txt
(3.63 KiB) Downloaded 54 times

6D Tigriduotorus (((II)I)((II)I)).txt
(5.4 KiB) Downloaded 56 times

6D Toratigritorus ((((II)I)(II))I).txt
(12.36 KiB) Downloaded 54 times

7D Spheritigriduotorus (((II)I)((II)I)I).txt
(1.77 KiB) Downloaded 58 times



9D Triotorus Tiger (((II)I)((II)I)((II)I)).txt
(1.78 KiB) Downloaded 60 times
in search of combinatorial objects of finite extent
ICN5D
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Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: CalcPlot3D

Postby ICN5D » Sun Jun 08, 2014 7:09 pm

* UPDATE *

This is a newer version of 7D Spheritigriduotorus , (((II)I)((II)I)I) , with the addition of a few more rotation equations.

7D Spheritigriduotorus (((II)I)((II)I)I).txt
(5.36 KiB) Downloaded 56 times


I noticed that after loading an explorer script, you still have to go back into View Settings > Advanced/Other Settings > # Rotation Steps > Other

and select OK , for the value to set properly. This should allow a smoother, easier to handle rotation of structures.


Then, play with the sliders to see the shape! Try different combinations of full left/right, then move only one slider at a time, and you will find some really cool stuff you never saw before. The sliders rotate a full 90 degrees at 0~1.57, but could be set for a full 180 or 360 with 3.14 or 7.28, respectively.
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: CalcPlot3D

Postby ICN5D » Wed Jun 25, 2014 12:08 am

Here's a list of all implicit equations I have made to explore toratopes in CalcPlot, in tabular form:

Code: Select all
nD Name - Notation , Trace-Array // Inflation Sequence // Diameter Hierarchy
------------------------------------------------------------------------------
Surface Equation
• 3D Translation Equations
• 3D Rotation Equations
• 3D Trans + Rot Equations
-- Notes




3D Torus - ((II)I) , (1)-Torus // circle-->circle // ((maj)min)
-----------------------------------------------------------------
(sqrt(x^2+y^2) - R1)^2 + z^2 - R2^2 = 0
• ((II)I)
(sqrt(x^2+y^2) - 2)^2 + z^2 - 0.5^2 = 0
• ((I)I)
(sqrt(x^2+a^2) - 2)^2 + y^2 - 0.5^2 = 0
• ((II))
(sqrt(x^2+y^2) - 2)^2 + a^2 - 0.5^2 = 0
• ((Iy)Y)
(sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 0.5^2 = 0



4D Glome (IIII) , 4-Ball
---------------------------------
x^2 + y^2 + z^2 + w^2 - R^2 = 0
• (IIIi)
x^2 + y^2 + z^2 + a^2 - 2^2 = 0
• (IYIy)
x^2 + (y*sin(a))^2 + z^2 + (y*cos(a))^2 - 2^2 = 0




4D Toripshere ((III)I) , 31-Torus // circle-->sphere // ((maj)min)
--------------------------------------------------------------------
(sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
• ((IIi)I) - torus
(sqrt(x^2 + y^2 + a^2) - 4)^2  + z^2 -0.75^2 = 0
• ((III)) - concentric spheres
(sqrt(x^2 + y^2 + z^2) - 4)^2  + a^2 -0.75^2 = 0
Ymin/max to -3 / 5
• ((IIz)Z) - rotation
(sqrt(x^2 + y^2 + (z*cos(a))^2) - 4)^2  + (z*sin(a))^2 -0.75^2 = 0
0 < a < 1.57
z= -2 / 5



4D Spheritorus ((II)II) , 22-Torus // sphere-->circle // ((maj)min)
---------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
• ((I)II) - displaced spheres
(sqrt(x^2 + a^2) - 2.5)^2 + y^2 + z^2 - 1^2 = 0
-4 < a < 4
• ((II)Ii) - torus
(sqrt(x^2 + y^2) - 2.5)^2 + z^2 + a^2 - 1^2 = 0
-0.7 < a < 0.7
• ((IY)Iy) - rotation
(sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2 + (y*cos(a))^2 - 1^2 = 0



4D Ditorus (((II)I)I) , 211-Ditorus // circle-->circle-->circle // (((maj)med)min)
-----------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0
• (((I)I)I) - displaced toruses
(sqrt((sqrt(x^2 + a^2) - 2.5)^2 + y^2) - 1)^2 + z^2 -0.5^2 = 0
-4.2 < a < 4.2
• (((II))I) - concentric toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + a^2) - 1)^2 + z^2 -0.5^2 = 0
-1.4 < a < 1.4
• (((II)I)) - cocircular toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + z^2) - 1)^2 + a^2 -0.5^2 = 0
-0.5 < a < 0.5
Zmin/max to  -1 / 5
Ymin/max to -3 / 5
• (((IY)y)I)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + z^2 = -0.5^2
• (((IY)I)y)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2) - 1)^2 + (y*cos(a))^2 = -0.5^2
• (((II)Z)z)
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + (z*sin(a))^2) - 1)^2 + (z*cos(a))^2 = -0.5^2



4D Tiger ((II)(II)) , (00)0-Tiger // circle-->duoring // ((maj1)(maj2)min)
---------------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2= 0
• ((II)(Ii)) - vertical stack of torii
(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0
-2.5 < a < 2.5
• ((IY)(Iy))
(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0
• ((IA)(Ia)) - Rotation + Translation
(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 -0.5^2 = 0
-- a translates
-- b rotates



5D (((II)I)(II)) - Tigritorus , (10)0-Tiger // circle-->duoring-->circle // (((maj1)med1)(med2)min)
---------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) -R1)^2 + z^2) -R2a)^2 + (sqrt(w^2 + v^2) -R2b)^2 -R3^2 = 0
• (((Ii)i)(II)) - 4 torii in 1x1x4 column
(Sqrt((Sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + z^2) -2)^2 -0.5^2 = 0
• (((II)i)(Ii)) - 4 torii in 2 concentric along 1x1x2 column
(Sqrt((Sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + (Sqrt(z^2 + b^2) - 2)^2 - 0.5^2 = 0
• (((Ii)I)(Ii)) - 4 torii in 2x1x2 vertical square
(Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + y^2) -1.2)^2 + (Sqrt(z^2 + b^2) -1.5)^2 -0.4^2 = 0
• (((IY)z)(Zy)) Rotation
(Sqrt((Sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (z*cos(b))^2) - 1.2)^2 + (Sqrt((z*sin(b))^2 + (y*cos(a))^2) - 1.5)^2 - 0.5^2 = 0
(Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + b^2) -1.2)^2 + (Sqrt(y^2 + z^2) - 1.5)^2 - 0.5^2 = 0



5D ((((II)I)I)I) - Tritorus , 2111-Tritorus // torus-->torus // ((((maj)sec)tert)min)
--------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 - R4^2= 0
• ((((I)I))I)
(sqrt((sqrt((sqrt(x^2 + a^2) - 9)^2 + y^2 ) - 4)^2 + b^2) - 2)^2 + z^2 - 1.25^2= 0
• ((((Iy)Y)z)Z)
(sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) - 9)^2 + (y*sin(a))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 - 1.25^2= 0
• ((((Iz)Y)y)Z)
(sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 9)^2 + (y*sin(b))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0
• ((((IY)z)y)Z)
(sqrt((sqrt((sqrt(x^2 + (y*sin(b))^2) - 9)^2 + (z*cos(a))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0
• ((((XY)x)I)y)
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 8)^2 + (x*cos(a))^2 ) - 4)^2 + z^2) - 2)^2 + (y*cos(b))^2 - 1.25^2= 0



5D (((II)(II))I) - Toritiger , (00)1-Tiger // circle-->circle-->duoring // (((maj1)(maj2)med)min)
---------------------------------------------------------------------------------------------------
((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 - R3^2 = 0
• (((I)(I))I)
((sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 - 1.2)^2 + z^2 - 0.75^2 = 0
• (((II)(I)))
((sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 - 1.2)^2 + b^2 - 0.75^2 = 0
simplified to : ((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2 - R3^2 = 0
rotation:
((sqrt(x^2 + (y*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 2.5)^2 - 1.2)^2 + (z*sin(b))^2 - 0.75^2 = 0



5D (((II)I)II) - Spheriditorus , 212-Ditorus // sphere-->circle-->circle // (((maj)med)min)
------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2 - R3^2 = 0
• (((I))II)
(sqrt((sqrt(x^2 + a^2) - 3.5)^2 + b^2 ) - 2)^2 + y^2 + z^2 = 1^2
(sqrt((sqrt(x^2 + (y*cos(a))^2) - 4)^2 + (z*cos(b))^2 ) - 2)^2 + (y*sin(a))^2 + (z*sin(b))^2 = 1^2
------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2 - R3^2 = 0
(sqrt((sqrt(x^2 + (y*cos(w))^2) - R1)^2 + (z*cos(v))^2 ) - R2)^2 + (y*sin(w))^2 + (z*sin(v))^2 - R3^2 = 0



5D (((III)I)I) - Ditorisphere , 311-Ditorus // circle-->circle-->sphere // (((maj)med)min)
----------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2))^2 - R3^2 = 0
major - R1 / medium - R2 / minor - R3
• (((III)I)I)
(sqrt((sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + (z*cos(b))^2 = .5^2



6D (((II)I)((II)I)) - Tigric Duotorus , (11)0-Tiger // tiger-->duoring  (((maj1)med1)((maj2)med2)min)
--------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3^2 = 0
• (((I)I)((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2
• (((Xz)Y)((Zx)y))
(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(c))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2
• (((Xz)Y)((Zy)x))
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(b))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(b))^2) - 2)^2 + (x*cos(c))^2) -1)^2 = 0.4^2
• (((II))((I)))
(sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2
• (((XY)z)((Zx)y))
(sqrt((sqrt((x*sin(b))^2 + (y*sin(c))^2) - 2)^2 + (z*cos(a))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2
-- A,C=1.57 / B=0.785 is (((xI))((xI))), quadruple tiger cage // A,C=0 / B=0.785 is (((x)I)((x)I)) 2x oblique tiger scan
• (((IA))((Ia))) translate A , rotate B
(sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) -1)^2 = 0.4^2
-- B=0.785 , Adjust A for flythrough of (((OI))((OI))) oblique structure, di-duoring structural scan with R5=0.2
-- XYZ = -5/+5
-- A = -4.5~4.5
• (((A)I)((Ca)c)) translate A,C  rotate B,D
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + (z*cos(d) - c*sin(d))^2) -1)^2 = 0.4^2
-- B=0.785 ; C,D=0 , Adjust A for fantastic diagonal translate along 2x2 square of tigers (((I)I)((I)I)) !!



6D ((((II)I)I)(II)) - Tigriditorus , (20)0-Tiger // circle-->duoring-->torus // ((((maj)sec)tert1)(tert2)min)
-------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3a)^2 + (sqrt(v^2 + u^2) - R3b)^2 - R4^2  = 0
• ((((I))I)(I)) - ((((X))Y)(Z))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4.5)^2 + b^2) - 2.2)^2 + y^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2  = 0
• ((((II)))(I)) - ((((XY)))(Z))
(sqrt((sqrt((sqrt(x^2 + y^2) - 4.5)^2 + a^2) - 2.2)^2 + b^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2  = 0
• ((((Xz)x)Y)(Zy))
(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4.5)^2 + (x*cos(b))^2) - 2.2)^2 + (y*sin(c))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((Xy)z)Y)(Zx))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.5)^2 + (z*cos(b))^2) - 2.2)^2 + (y*sin(a))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((Xz)y)Y)(Zx))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 4.5)^2 + (y*cos(b))^2) - 2.2)^2 + (y*sin(b))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((XY)y)z)(Zx)) same as ((((Xz)x)Y)(Zy))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(a))^2) - 4.5)^2 + (y*cos(a))^2) - 2.2)^2 + (z*cos(b))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
Xmin/max= -9,+9
Ymin/max= -9,+9
Zmin/max= -9,+9



6D (((II)(II))(II)) -  Double Tiger , ((00)0,0)0-Tiger //  torus-->trioring  (((maj1)(maj2)med)(maj3)min)
-----------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3^2 = 0
• (((I)(I))(I))
(sqrt((sqrt(x^2 + a^2) - R1)^2 + (sqrt(y^2 + b^2) - R2)^2) - R3)^2 + (sqrt(z^2 + c^2) - R4)^2 - R5^2 = 0
• (((I)(I))(I))
(sqrt((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + c^2) - 3)^2 - 0.5^2 = 0
• (((Xz)(Yx))(Zy))
(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2.5)^2 - 0.5^2 = 0
Octatangent Cut at
a = 0.36497
b = 0.3492
c = 0.91242
• (((Xy)(Yz))(Zx))
(sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 0.5^2 = 0
Xmin/max= -9,+9
Ymin/max= -9,+9
Zmin/max= -9,+9



6D ((II)(II)(II)) - Tritiger , Spheritriotorus // (000)-Tiger // sphere-->trioring // ((maj1)(maj2)(maj3)min)
--------------------------------------------------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2^2 = 0
• ((I)(I)(I)) - 2x2x2 array of 8 spheres
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 - 1^2 = 0
• ((Xy)(Yz)(Zx))
(sqrt((x*sin(c))^2 + (y*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 1^2 = 0
XYZmin/max = -5,+5
• ((Xz)(Yx)(Zy))
(sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 3)^2 - 0.8805^2 = 0
-- Special diameter values for 45x45x45 cut, dodecatangent



6D ((((II)(II))I)I) - Ditoratiger // (00)2-Tiger // ditorus-->duoring // ((((maj1)(maj2)sec)tert)min)
-----------------------------------------------------------------------------------------------------
(sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0
• ((((X)(Y)))Z)
(sqrt(((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2 - 3)^2 + c^2) - 1.75)^2 + z^2 - 0.75^2 = 0
• ((((Xz)(Yx))y)Z)
(sqrt(((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2 - 3)^2 + (y*cos(c))^2) - 1.75)^2 + (z*sin(a))^2 - 0.75^2 = 0
• ((((Xy)(Yz))x)Z)
(sqrt(((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 3)^2 - 3)^2 + (x*cos(c))^2) - 1.75)^2 + (z*sin(b))^2 - 0.75^2 = 0



6D (((II)I)(II)I) - Spheritigritorus , (10)0-Tiger , sphere-->duoring-->circle // (((maj)med1)(med2)min)
------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 + u^2 - R3^2 = 0
• (((Ii)i)(Ii)I) = (((I))(I)I) - 4x2 array of 8 spheres
(sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0
• (((II)i)(Ii)i) = (((II))(I)) - 2 conc stacked 2 high of 4 torii
(sqrt((sqrt(x^2 + y^2) -2)^2 + a^2) -1)^2 + (Sqrt(z^2 + b^2) -2)^2 + c^2 -0.75^2 = 0
• (((Ii)i)(II)i) = (((I))(II)) - vert column of 4 torii
(sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0
• (((xz)y)(yx)z)
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0
• (((xy)z)(yx)z)
(sqrt((sqrt((x*sin(c))^2 + (y*cos(b))^2) -2)^2 + (z*cos(a))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0



6D ((((II)I)(II))I) - Toratigritorus , (10)1-Tiger // circle-->duoring-->circle // ((((maj)sec1)(sec2)tert)min)
----------------------------------------------------------------------------------------------------------------
((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3)^2 + u^2 - R4^2 = 0
• ((((I))(I))I) - 4x2 array of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + c^2) - 2.25)^2 - 1.75)^2 + z^2 - 1^2 = 0
• ((((II))(I))) - 2 cocirc by 2 conc stacked 2 high of 8 torii
((sqrt((sqrt(x^2 + y^2) - 3.75)^2 + a^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0
• ((((I))(II))) - 2 cocirc stacked 4 high of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + z^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0
• ((((I)I)(I))) - 2 cocirc in 2x1x2 vert square of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + y^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0
• ((((Xy)z)(Yx))Z)
((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3.75)^2 + (z*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (x*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(b))^2 - 1^2 = 0
• ((((Xz)y)(Yx))Z)
((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 3.75)^2 + (y*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(a))^2 - 1^2 = 0
• ((((Xy)x)(Yz))Z)
((sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) - 3.75)^2 + (x*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(c))^2 - 1^2 = 0
• ((((Ac)I)(a))C)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 3.75)^2 + y^2) - 1.9)^2 + (sqrt(0^2 + (x*cos(b) - a*sin(b))^2) - 2.25)^2 - 1.75)^2 + (z*sin(d) + c*cos(d))^2 - 1^2 = 0
-- Very good exploration of ((((I)I)(I))I) cut, though Z rotations make major-->minor concentric morph
• ((((A)C)(ac))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 3.75)^2 + (y*sin(d) + c*cos(d))^2) - 1.9)^2 + (sqrt((x*cos(b) - a*sin(b))^2 + (y*cos(d) - c*sin(d))^2) - 2.25)^2 - 1.75)^2 + z^2 - 1^2 = 0
-- A= -1.29 , C= 0.9 : Adjust B,D angles to all four combinations, very cool stuff!



7D ((((II)(II))I)(II)) - Tigritiger , ((00)1,0)0 - Tiger   //   tiger-->tiger  ,  ditorus-->trioring  // ((((maj1)(maj2)sec)tert)(maj3)min)
------------------------------------------------------------------------------------------------------------------------------
(sqrt(((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2)^2 + v^2) - R3)^2 + (sqrt(u^2 + t^2) - R1c)^2 - R4^2 = 0
• ((((I)(I)))(I)) , ((((Xa)(Yb))c)(Zd))
(sqrt(((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2 - 3.75)^2 + d^2) - 2)^2 + (sqrt(z^2 + c^2) - 4)^2 - 1^2 = 0
-- XYZ = -6,+6 / **Zscale=0.5**
• ((((Xa)(Yz))y)(Zx))
(sqrt(((sqrt((x*sin(d))^2 + a^2) - 3)^2 + (sqrt((y*sin(c))^2 + (z*cos(b))^2) - 3)^2 - 3.75)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((z*sin(b))^2 + (x*cos(d))^2) - 4)^2 - 1^2 = 0
• ((((Xa)(Yz))x)(Zy))
(sqrt(((sqrt((x*sin(c))^2 + a^2) - 3)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 3)^2 - 3.75)^2 + (x*cos(c))^2) - 2)^2 + (sqrt((z*sin(b))^2 + (y*cos(d))^2) - 4)^2 - 1^2 = 0
-- ETEs scan past horiz sqr of 4 tigritoruses ((((I)(I))I)(II)) or vert sqr of 4 tritoruses ((((II)(I))I)(I))
-- Set B,C,D=0 and A=4 to translate to 2x (((II)I)(II)) , then rotate C to transform into 2x ((((II)I)I)I)
-- XYZ = -6,+6 / **Zscale=0.5**



7D (((II)I)((II)I)I) - Spheritiger Duotorus , (110)0-Tiger // sphere-->duoring[maj1,2]-->duoring // (((maj1)med1)((maj2)med2)min)
-----------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 + t^2 - R3^2 = 0
• (((I))((I))I)
(sqrt((sqrt(x^2 + a^2) - 2.35)^2 + b^2) -1.15)^2 + (sqrt((sqrt(y^2 + c^2) - 2.6)^2 + d^2) -1.25)^2 + z^2 - 0.5^2 = 0
• (((Xz)y)((Yx)d)Z)
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2.35)^2 + (y*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.6)^2 + d^2) -1.25)^2 + (z*sin(a))^2 - 0.5^2 = 0
• (((Xy)z)((Yc)x)Z)
(sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 2.35)^2 + (z*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(a))^2 + c^2) - 2.6)^2 + (x*cos(d))^2) -1.25)^2 + (z*sin(b))^2 - 0.5^2 = 0
• (((Xa)x)((Yz)y)Z)
(sqrt((sqrt((x*sin(b))^2 + a^2) - 2.35)^2 + (x*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(d))^2 + (z*cos(c))^2) - 2.6)^2 + (y*cos(d))^2) -1.25)^2 + (z*sin(c))^2 - 0.5^2 = 0



8D ((II)(II)(II)(II)) - Tetratiger , (0000)0-Tiger // glome-->quattroring // ((maj1)(maj2)(maj3)(maj4)min)
------------------------------------------------------------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 + (sqrt(t^2 + s^2) - R1d)^2- R2^2 = 0
• ((I)(I)(I)(I)) - 2x2x2x2 array of 16 glomes
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 + (sqrt(d^2 + 0^2) - 2.5)^2 - 1^2 = 0
• ((Ia)(Ib)(I)(Cd))
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + 0^2) - 2.5)^2 + (sqrt(c^2 + d^2) - 2.5)^2 - 1^2 = 0
-- Adjusting C translates away from center of 2x2x2x2 array of glomes, passing by either +W or -W 2x2x2 cube array
-- Adjusting D controls merging of +W,-W 2x2x2 arrays in 4-space
• ((Xc)(Yz)(Zx)(Cy))
(sqrt((x*sin(0))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (y*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Set C=3.5 for only visible morphings.
• ((Xc)(Yx)(Zy)(Cz))
(sqrt((x*sin(b))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(0))^2 + (x*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(d))^2 + (y*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (z*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Adjusting B scans along diagonal 2x2x2 array, max illum at 0.785/45deg
-- Adjusting C translates away fr center of 2x2x2x2 array, max illum at +2.5,-2.5 in 4-space



8D ((((II)I)(II))((II)I)) - Tigritorus Duotoric Torus , ((10)0,1)0-Tiger , circle-->duoring-->circle-->duoring-->circle , ((((maj1)sec1)(sec2)tert)((maj2)sec3)min)
---------------------------------------------------------------------------------------------------------------------
((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3)^2 + (sqrt((sqrt(u^2 + t^2) - R1b)^2 + s^2) - R2c)^2 - R4^2 = 0
• ((((I))(I))((I)))
((sqrt((sqrt(x^2 + a^2) - 5)^2 + 0^2) - 2.25)^2 + (sqrt(y^2 + b^2) - 2.25)^2 - 1.8)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.75^2 = 0



9D (((II)I)((II)I)((II)I)) - Tritigric Triotorus , (111)0-Tiger // sphere-->trioring-->trioring // (((maj1)med1)((maj2)med2)((maj3)med3)min)
----------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 + (sqrt((sqrt(t^2 + s^2) - R1c)^2 + r^2) - R2c)^2 - R3^2 = 0
• (((I))((I))((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(y^2 + b^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.5^2 = 0



9D (((((II)I)(II))I)((II)I)) - Tigritoric Tigritorus, ((10)1,1)0-Tiger, circle-->duoring-->circle-->tiger-->circle
--------------------------------------------------------------------------------------------------------------------
(sqrt(((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R4)^2 + u^2) - R5)^2 + (sqrt((sqrt(t^2 + s^2) - R2c)^2 + r^2) - R7)^2 - R8^2 = 0

(sqrt(((sqrt((sqrt(x^2 + y^2) - 5.5)^2 + z^2) - 2.75)^2 + (sqrt(w^2 + v^2) - 3.5)^2 - 3)^2 + u^2) - 1.9)^2 + (sqrt((sqrt(t^2 + s^2) - 2..5)^2 + r^2) - 1.25)^2 - 0.75^2 = 0
• (((((I))(I)))((I)))
(sqrt(((sqrt((sqrt(x^2 + a^2) - 5.5)^2 + 0^2) - 2.75)^2 + (sqrt(y^2 + b^2) - 3.5)^2 - 3)^2 + c^2) - 1.9)^2 + (sqrt((sqrt(z^2 + d^2) - 2.5)^2 + 0^2) - 1.25)^2  - 0.75^2 = 0
in search of combinatorial objects of finite extent
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Re: CalcPlot3D

Postby ICN5D » Wed Jul 23, 2014 10:55 pm

Here is a 2D exploration of the Ditorus (((II)I)I) , T^3

CalcPlot Script:

Exploring T^3 in 2D Slices.txt
(8.56 KiB) Downloaded 58 times

- Render in 30-35 cubes, move the translate sliders slowly ( for certain computers that are 8 yrs old and broken, like mine )
Bounding Box:
- Set X,Y to -5,+5
- Set Z to -0.01 , +0.01 for 2D
- Go To View Settings > Advanced/Other Settings > # Rotation Steps > Other and click [OK] for 180 rotate steps


Equations Used in Script, by Steps :

(((II)I)I) Exploring in 2D Slices
-------------------------------------------------------
• Step 1: (((Ac)C)a) - Rotate + Translate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (y*sin(d) + c*cos(d))^2) - 1)^2 + (x*cos(b) - a*sin(b))^2 -0.5^2 = 0
- A,C Translates
- B,D Rotates

• Step 2: (((Ac)a)C) - Rotate + Translate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (y*sin(d) + c*cos(d))^2 -0.5^2 = 0
-- I think this one is identical to above ...

• Step 3: (((I))I) - 4x1 row of circles
(sqrt((sqrt(x^2 + a^2) - 2)^2 + b^2) - 1)^2 + y^2 -0.5^2 = 0
- A,B Translates away from center

• Step 4: (((II))) - 4x concentric circles
(sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + b^2 -0.5^2 = 0

• Step 5: (((I)I)) - 2x pair in 2x1 row of circles
(sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) - 1)^2 + b^2 -0.5^2 = 0
in search of combinatorial objects of finite extent
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Re: CalcPlot3D

Postby ICN5D » Sun Sep 28, 2014 5:43 pm

Here is the updated list of explore functions for graphing toratope intercepts. This includes many new ones, esp in 5D and 7D. All in all, it looks like I've surveyed 28 of them so far. This is not a CalcPlot script, just a table of functions.

VVV See Further Below for Updated File VVV
Last edited by ICN5D on Sun Sep 06, 2015 10:57 pm, edited 2 times in total.
in search of combinatorial objects of finite extent
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Re: CalcPlot3D

Postby ICN5D » Thu Jan 08, 2015 11:55 pm

A new version of my Hypertorus Explore Function Library, updated to 1/8/2015. There are a lot of new 5D and 7D toratopes, plus a 10D, with new explore function types.


VVV See Further Below for Updated File VVV
Last edited by ICN5D on Sun Sep 06, 2015 10:59 pm, edited 2 times in total.
in search of combinatorial objects of finite extent
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Re: CalcPlot3D

Postby ICN5D » Fri Feb 13, 2015 12:26 am

Some new exploration scripts:

5D Conindrone Script.txt
(2.17 KiB) Downloaded 55 times


Cyltrianglinder Animation Script.txt
(3.76 KiB) Downloaded 39 times


4D Bicircular Tegum.txt
(1.77 KiB) Downloaded 42 times


4D Duocylinder (II)(II) , IOIO.txt
(1.8 KiB) Downloaded 44 times


7D ((((II)(II))I)(II)) Tigritiger.txt
(3.53 KiB) Downloaded 48 times


7D ((((II)I)(II))(II)) Double Tiger Torus.txt
(5.34 KiB) Downloaded 46 times


7D Double Tiger 1C-Torus (((II)(II))((II)I)).txt
(5.45 KiB) Downloaded 42 times


5D Torispheritorus (((II)II)I).txt
(1.74 KiB) Downloaded 41 times


5D Ditorisphere (((III)I)I).txt
(1.74 KiB) Downloaded 42 times


5D Spheriditorus (((II)I)II).txt
(1.74 KiB) Downloaded 44 times
in search of combinatorial objects of finite extent
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Re: CalcPlot3D

Postby ICN5D » Tue Feb 17, 2015 12:55 am

Some new toratopes I explored recently.

Code: Select all
7D ((((II)(II))I)(II)) - Tigritiger , T3xC3 = S1xC2xS1xC2 = S1x[(T2xC2)*S1] , ((((maj1)(maj2)sec)tert)(maj3)min)
-------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2 = Rminor^2
• ((((I)(I)))(I)) : 2x2x2x[R1 pair] array of 16 tori
(sqrt((sqrt((sqrt(x^2+0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1
———— XYZbox = -17/+17
• ((((A)(Ic))a)(C))
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -7.5)^2 + (sqrt(y^2+(z*cos(d)-c*sin(d))^2) -7.5)^2) -3.5)^2 +(x*cos(b)-a*sin(b))^2) -1.5)^2 + (sqrt((z*sin(d)+c*cos(d))^2+0^2) -3.5)^2 = 1
——— -15 < a,c < 15
——— 0 < b,d < 1.5707
• ((((Xc)(I))a)(Cb))
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+(z*cos(d) - c*sin(d))^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(a))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(b))^2) -3.5)^2 = 1
— -15 < c < 15
— 0 < a,b,d < 1.5707
• ((((X)(Ia))c)(Cb))
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+0^2) -7.5)^2 + (sqrt(y^2+(x*cos(a))^2) -7.5)^2) -3.5)^2 +(z*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(a))^2) -3.5)^2 = 1
• ((((A)(Ca))c)(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +(y*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1
——— -17 < a,c < 17
——— 0 < b,d < 1.5707
• ((((A)(Ca)))(Ic)) ** good function **
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1
——— -17 < a,c < 17
——— 0 < b,d < 1.5707
• ((((A)(C))a)(Ic))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(b) - a*sin(b))^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1
——- a=7.5,c=7.5±3.5 for 1x1x2 column of tori in cut (((()())I)(II))




7D (((((II)(II))I)I)I) :  Tritoratiger : T4xC2 , (((((R1a)(R1b)R2)R3)R4)minor)
------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2

(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2
• Diameter Adjustment Equation
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 +z^2 = 1
• (((((I)(I))))I) - 2x2x[R1quartet] array of 16 tori
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 + (sqrt(y^2+0^2) -15)^2) -7)^2 +0^2) -3.2)^2 +0^2) -1.6)^2 +z^2 = 1
—— XYZbox = -35 / +35
—— 55 cubes
• (((((A)(Ia))c)d)Z) : X -> A,a  /  Z -> c,d
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -15)^2 + (sqrt(y^2+(x*cos(b) - a*sin(b))^2) -15)^2) -7)^2 +(z*cos(c))^2) -3.2)^2 +(z*cos(d))^2) -1.6)^2 +(z*((sin(c))*(sin(d))))^2 = 1
—— -35 < a < 35
—— 0 < b,c,d < 1.5707
• (((((Xc)(Ix))b)a)Z) : X -> x  /  Z -> a,b,c
(sqrt((sqrt((sqrt((sqrt((x*sin(d))^2+(z*cos(c))^2) -15)^2 + (sqrt(y^2+(x*cos(d))^2) -15)^2) -7)^2 +(z*cos(b))^2) -3.2)^2 +(z*cos(a))^2) -1.6)^2 +(z*((sin(a))*(sin(b))*(sin(c))))^2 = 1
—— 0 < a,b,c,d < 1.5707





8D ((((II)I)I)(((II)I)I)) : Duoditorus Tiger , S1xC2xC2xC2
-------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 +w^2) -R3a)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -R1b)^2 +t^2) -R2b)^2 +s^2) -R3b)^2 = Rminor^2

(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +z^2) -5)^2 +w^2) -2.5)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -10)^2 +t^2) -5)^2 +s^2) -2.5)^2 = 0.5
• ((((II)))(((I))))
(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+0^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
— XYZbox = -27 / +27
• ((((IA)))(((Ia)))) : y -> A,a
(sqrt((sqrt((sqrt(x^2+(y*sin(b) + a*cos(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
• ((((AY)c)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*((sin(c))*(sin(d))))^2) -10)^2 +(y*cos(c))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*sin(b) + a*cos(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
• ((((AY)c)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*((sin(c))*(sin(d))))^2) -10)^2 +(y*cos(c))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5




8D ((((II)(II))I)((II)I)) : [Toratiger-Torus] Tiger , S1xC2xS1xS1xC2 , T2xC2xC3 , S1xC2xS1xC3
------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R3)^2 +v^2) -R4)^2 + (sqrt((sqrt(u^2+t^2) -R1c)^2 +s^2) -R2)^2 = Rm^2
• Trace Array Diameter Adjustment Equation
(sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) -b)^2 +0^2) -(b/2))^2 = 1
• ((((I)(I)))((I))) : 2x2x4x[R1 pair] array of 32 toruses
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— XYZbox = -22 / +22
• ((((I)(I)))((I))) : standard explore function slate
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
• ((((A)(Ca)))((Ic))) : X -> A,a / Y -> C,c
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+(y*cos(d) - c*sin(d))^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((Ic)(A))a)((C))) : Y -> A,a / Z -> C,c
(sqrt((sqrt((sqrt(x^2+(z*cos(d) - c*sin(d))^2) -9)^2 + (sqrt((y*sin(b) + a*cos(b))^2+0^2) -9)^2) -4.5)^2 +(y*cos(b) - a*sin(b))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((A)(I))c)((C)a)) : X -> A,a / Z -> C,c
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +(z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +(x*cos(b) - a*sin(b))^2) -3.25)^2 = 1





9D (((II)I)((II)I)((II)I)) - Triger Triotorus , S2xC3xC3 = S2x[T2*T2*T2] , (((maj1)med1)((maj2)med2)((maj3)med3)min)
------------------------------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2c)^2 = R3^2
• (((I))((I))((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(y^2 + b^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.5^2 = 0
• (((A))((Ca)c)((I)))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + 0^2) -2)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((sqrt(z^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1
• (((A)c)((Ia))((c)))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt(y^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + 0^2) - 2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1




9D (((((II)I)(II))I)((II)I)) - [ToratigerTorus-Torus] Tiger , T3xC2xC3 = S1xC2xT2xC2xS1 , (((((R1a)R2a)(R1b)R3)R4)((R1c)R2b)Rm)
--------------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2b)^2 = Rminor^2
• Diameter Adjustment Equation for Trace Array
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) - b)^2 +0^2) - (b/2))^2 = 1
--- a=15
--- b=7
--- c=4
--- d=1.75
• (((((I))(I)))((I))) : 4x2x4x[R1 pair] of 64 Toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -10)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1
—— XYZbox = -32 / +32
—— 55 cubes






9D ((((((II)I)(II))I)I)(II)) : [DitoratigerTorus-Circle] Tiger , S1xC2xT2xC2xS1
--------------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2) -R5)^2 + (sqrt(s^2+r^2) -R1c)^2 = Rminor^2
• ((((((I))(I))))(I)) Diameter Adjustment Function
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 + (sqrt(z^2+0^2) -(a/5))^2 = 1
--- a=27 , b=7.1 , c=3.2 , d=1.6
--- XYbox = -55 / +55 , Zbox = -25 / +25


in search of combinatorial objects of finite extent
ICN5D
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Re: CalcPlot3D

Postby ICN5D » Thu Feb 26, 2015 2:25 am

Some new shapes to add, in light of recent exploration:

Code: Select all
7D (((((II)I)(II))I)I) : Ditoratiger Torus, T3xC2xS1
---------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2

• (((((I))(I)))I) : 16x Tori in 4x2x[R1 quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +z^2 = 1

• (((((I))(I))I)) : 16x Tori in 4x2x[R1 pair]x[Rm pair]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +z^2) -1.75)^2 +0^2 = 1

• (((((I))(II)))) : 16x Tori in 1x1x4x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+z^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1

• (((((I)I)(I)))) : 16x Tori in 2x1x2x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +y^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1

• (((((II))(I)))) : 16x Tori in 1x1x2x[R1 pair]x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -14)^2 +0^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1

• Diameter Adjustment Equation of (((((I))(I)))I)
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt(y^2+0^2) -b)^2) -c)^2 +0^2) -d)^2 +z^2 = 1

a=14 ; b=7 ; c=3.5 ; d=1.75
--- XYZbox = -30,+30



And, a new frontier, nine dimensional explore functions of a 4D intercept array! This is the first time I explored a 9D toratope this deeply, in addition to being a 4D intercept array. I was very curious as to what morphs it would take on visually, and had to check it out:

Code: Select all
9D (((((II)I)(II))(II))(II)) : Triple Tiger 1A-Torus , T3xC4xS1
------------------------------------------------------------------
(II) - S1
((II)I) - T2
(((II)I)I) - T3
((((II)I)I)I) - T4
((((II)(II))I)I) - T3xC2
((((II)(II))(II))I) - T3xC3
((((II)(II))(II))(II)) - T3xC4
(((((II)I)(II))(II))(II)) - T3xC4xS1
--------------------------------------
(((((II)I)(II))(II))(II))
((((II)I)(II))(II))(II)
( ( ( (II) I) (II)) (II)) (II)
( ( ( (xy) z) (wv)) (ut)) (sr)
( ( ( (x+y) +z) + (w+v)) + (u+t)) + (s+r)
( ( ( (x+y -R1a) +z -R2) + (w+v -R1b) -R3) + (u+t -R1c) -R4) + (s+r -R1d) = Rminor
( ( ( (x+y -R1a)² +z -R2)² + (w+v -R1b)² -R3)² + (u+t -R1c)² -R4)² + (s+r -R1d)² = Rminor²
( ( ( (√(x+y) -R1a)² +z -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( ( ( √((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( ( √((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( √((√((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)²) -R4)² + (√(s+r) -R1d)² = Rminor²
(√((√((√((√(x²+y²) -R1a)² +z²) -R2)² + (√(w²+v²) -R1b)²) -R3)² + (√(u²+t²) -R1c)²) -R4)² + (√(s²+r²) -R1d)² = Rminor²
----------------------------------------------------------------------------------------------------------------------
• (√((√((√((√(x²+y²)-R1a)²+z²)-R2)²+(√(w²+v²)-R1b)²)-R3)²+(√(u²+t²)-R1c)²)-R4)²+(√(s²+r²)-R1d)² = Rminor²
• (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2) -R4)^2 + (sqrt(s^2+r^2) -R1d)^2 = Rminor^2
------------------------------------
(((((R1a)R2)(R1b)R3)(R1c)R4)(R1d)Rm)
(((((24)12)(12)6)(8)3)(4)1) - ring torus diameter values
• (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -24)^2 +z^2) -12)^2 + (sqrt(w^2+v^2) -12)^2) -6)^2 + (sqrt(u^2+t^2) -12)^2) -3)^2 + (sqrt(s^2+r^2) -6)^2 = 1
------------------
XYrange = -48,+48
Zrange = -15,+15
------------------
3D Scans of the (((((I))(I))(I))(I)) 4x2x2x2 Array, Adjust ‘a’ to move ± in 4-space by [±Rn] value
----------------------------------------------------------------------------------------------------
• ((((())(I))(I))(I)) : [±R1A±R2] intercepts are 4 places of 4x2x2 array of 16 toruses
(sqrt((sqrt((sqrt((sqrt(0^2+a^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48

• (((((I))())(I))(I)) : [±R1B] intercepts are 2 places of 8x2x2 array of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(a^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1

• (((((I))(I))())(I)) : [±R1C] intercepts are 2 places of 4x2x2x[R1 pair] of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(a^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1

• (((((I))(I))(I))()) : [±R1D] intercepts are 2 places of 4x2x2x[Rm pair] of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt(a^2+0^2) -4)^2 = 1
-----------------------------------------------
3D Explore Functions , Single Translate+Rotate
-----------------------------------------------
• (((((a))(A))(I))(I)) : Single Trans+Rotate; {±R1A±R2 of 4x2x2} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(0^2+(x*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57

• (((((I))(a))(A))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -12)^2) -6)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -24 < a < 24
--- 0 < b  <1.57

• (((((I))(I))(A))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1C of 4x2x2x[R1-pair]}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt((z*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1
--- -13 < a < 13
--- 0 < b  <1.57

• (((((a))(I))(I))(A)) : {±R1A±R2 of 4x2x2} to {±R1D of 4x2x2x[Rm-pair]}
(sqrt((sqrt((sqrt((sqrt(0^2+(z*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57

• (((((A))(I))(a))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1A±R2 of 4x2x2}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57
--- Zrange = -15,15

• (((((I))(A))(I))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1
--- -24 < a < 24
--- 0 < b  <1.57
--- Zrange = -15,15
----------------------
Dual Translate+Rotate
----------------------
• (((((a))(Ac))(C))(I)) : Dual T+R , explores 4x2 dual-void rectangle array of ((((())(II))())(I))
(sqrt((sqrt((sqrt((sqrt((x*cos(b)-a*sin(b))^2) -24)^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+(y*cos(d)-c*sin(d))^2) -12)^2) -6)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1
--- [a,b,c,d] = [±24±12 ; 0 ; ±8 ; 1.57] : ((((())(II))())(I)) : [±R1A±R2]x[±R1C] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet]

• (((((A)c)(C))(a))(I)) : Dual T+R
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2 +(y*cos(d)-c*sin(d))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1

• (((((A))(Ca))(-))(Ic)) : Dual T+R, Preset [±R1C]=8, explores multiple hypervoids
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1

• (((((-)A)(Ca))(-))(Ic)) : Dual T+R, Preset Void Locations [±R1A]=24;[±R1C]=8, explores dual-hypervoid intercept region
(sqrt((sqrt((sqrt((sqrt(24^2) -24)^2 +(x*sin(b)+a*cos(b))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1
----------------------------------------------------------------
• Dual-Empty Hypervoid Intercepts <[For Navigational Purposes]>
(((((R1a)R2)(R1b))(R1c))(R1d))
(((((±24)±12)(±12))(±8))(±4))
------------------------------
(((((a)b)(c))(d))(e)) - [±a]x[±b]x[±c]x[±d]x[±e] : Hypervoid Locations of Ring Intercept Activity
-------------------------------------------------------------------------------------------------
(((((I)I)(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((I)I)))(I)), 2x1x2x[R1-quartet] of 16 tori
(((((a))(II))(b))(I)) - [±24±12]x[±8] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet] of 8 tori
(((((a))(I))(b))(II)) - [±24±12]x[±8] are 4x2 places of ((((I)))(II)), 1x1x8 column of 8 tori
(((((I))(a))(b))(II)) - [±12]x[±8] are 2x2 places of (((((I))))(II)), 1x1x16 column of 16 tori
(((((II))(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((II))))(I)), 1x1x2x[R1-octet] of 16 tori
(((((a)I)(b))(c))(II)) - [±24]x[±12]x[±8] are 2x2x2 places of ((((I)))(II)), 1x1x8 column of 8 tori
(((((a)I)(b))(I))(I)) - [±24]x[±12] are 2x2 places of ((((I))(I))(I)), 4x2x2 array of 16 tori
(((((a)I)(I))(b))(I)) - [±24]x[±8] are 2x2 places of ((((I)(I)))(I)), 2x2x2x[R1-pair] of 16 tori

in search of combinatorial objects of finite extent
ICN5D
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Re: CalcPlot3D

Postby ICN5D » Sun Mar 15, 2015 10:06 pm

**A new update of my Hypertorus Explore Function library **

This one has all of the corrected functions that were missing the square root, leading to squashed bands instead of round tori. This is the most correct and complete, with new explore functions for 9D toratopes.


Download the text file: ***Updated 3/15/2015***

Hypertorus Explore Function Library.txt
(72.09 KiB) Downloaded 38 times


Copy-Paste version, to use with CalcPlot, if you want to:

Code: Select all

*************************************************************************
nD Notation - Name , Fiber Bundle Sequence , Diameter Size Hierarchy   **
-----------------------------------------------------------------------**
Surface Equation                                                       **
• Translation Equations {a,b,c,d}                                      **
• Rotation Equations {X,x} {Y,y} {Z,z}                                 **
• Trans + Rot Equations {A,a} {C,c}                                    **
• MultiPosition Rotate Eq {X,a,b} {Y,a,b} {Z,a,b}                      **
• MultiDimension Rotate Eq {X,Y,[xy]} {X,Z,[xz]} {Y,Z,[yz]}            **
--- Notes                                                              **
                                                                       **
*************************************************************************
*************************************************************************
* Explore Function Library for Hypertoric Varieties *
*****************************************************


3D ((II)I) - Torus , S1xS1 , ((maj)min)
-----------------------------------------------------------------
(sqrt(x^2+y^2) - R1)^2 + z^2 - R2^2 = 0
• ((II)I)
(sqrt(x^2+y^2) - 2)^2 + z^2 - 0.5^2 = 0
• ((I)I)
(sqrt(x^2+a^2) - 2)^2 + y^2 - 0.5^2 = 0
• ((II))
(sqrt(x^2+y^2) - 2)^2 + a^2 - 0.5^2 = 0
• ((Iy)Y)
(sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 0.5^2 = 0
• ((II)I) 3D rotate
(sqrt((x*sin(a) + z*cos(a))^2+y^2) - 2)^2 + (x*cos(a) - z*sin(a))^2 - 0.5^2 = 0
• Rotate + Translate
(sqrt((x*sin(b) + (z+a)*cos(b))^2+y^2) - 2)^2 + (x*cos(b) - (z+a)*sin(b))^2 = 1





4D (IIII) - Glome , S3
---------------------------------
x^2 + y^2 + z^2 + w^2 - R^2 = 0
• (IIIi)
x^2 + y^2 + z^2 + a^2 - 2^2 = 0
• (IYIy)
x^2 + (y*sin(a))^2 + z^2 + (y*cos(a))^2 - 2^2 = 0





4D ((III)I) - Toripshere , S1xS2 , ((maj)min)
--------------------------------------------------------------------
(sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
• ((IIi)I) - torus
(sqrt(x^2 + y^2 + a^2) - 4)^2  + z^2 -0.75^2 = 0
• ((III)) - concentric spheres
(sqrt(x^2 + y^2 + z^2) - 4)^2  + a^2 -0.75^2 = 0
Ymin/max to -3 / 5
• ((IIz)Z) - rotation
(sqrt(x^2 + y^2 + (z*cos(a))^2) - 4)^2  + (z*sin(a))^2 -0.75^2 = 0
0 < a < 1.57
z= -2 / 5

• ((IIa)A) - Trans+Rotate
(sqrt(x^2 + y^2 + (z*cos(b/57.3)-a*sin(b/57.3))^2) - 3)^2  + (z*sin(b/57.3)+a*cos(b/57.3))^2 = 1





4D ((II)II) - Spheritorus , S2xS1 , ((maj)min)
---------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
• ((I)II) - displaced spheres
(sqrt(x^2 + a^2) - 2.5)^2 + y^2 + z^2 - 1^2 = 0
-4 < a < 4
• ((II)Ii) - torus
(sqrt(x^2 + y^2) - 2.5)^2 + z^2 + a^2 - 1^2 = 0
-0.7 < a < 0.7
• ((IY)Iy) - rotation
(sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2 + (y*cos(a))^2 - 1^2 = 0

• ((IA)Ia) - Trans+Rotate
(sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 3)^2 + z^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2 = 1






4D (((II)I)I) - Ditorus , T3 , (((maj)med)min)
-----------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0
• (((I)I)I) - displaced toruses
(sqrt((sqrt(x^2 + a^2) - 2.5)^2 + y^2) - 1)^2 + z^2 -0.5^2 = 0
-4.2 < a < 4.2
• (((II))I) - concentric toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + a^2) - 1)^2 + z^2 -0.5^2 = 0
-1.4 < a < 1.4
• (((II)I)) - cocircular toruses
(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + z^2) - 1)^2 + a^2 -0.5^2 = 0
-0.5 < a < 0.5
Zmin/max to  -1 / 5
Ymin/max to -3 / 5
• (((IY)y)I)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 9)^2 + (y*cos(a))^2) - 3)^2 + z^2 = 1
• (((IY)I)y)
(sqrt((sqrt(x^2 + (y*sin(a))^2) - 9)^2 + z^2) - 3)^2 + (y*cos(a))^2 = 1
• (((II)Z)z)
(sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*sin(a))^2) - 3)^2 + (z*cos(a))^2 = 1

Trans+Rotate:
• (((IA)a)I)
(sqrt((sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 6)^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2) - 3)^2 + z^2 = 1
• (((IA)I)a)
(sqrt((sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 6)^2 + z^2) - 3)^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2 = 1
• (((II)A)a)
(sqrt((sqrt(x^2 + y^2) - 6)^2 + (z*sin(b/57.3)+a*cos(b/57.3))^2) - 3)^2 + (z*cos(b/57.3)-a*sin(b/57.3))^2 = 1

(((II)I)I) Exploring in 2D Slices
------------------------------------
• Step 1: (((Ac)C)a) - Rotate + Translate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (y*sin(d) + c*cos(d))^2) - 1)^2 + (x*cos(b) - a*sin(b))^2 -0.5^2 = 0
- A,C Translates
- B,D Rotates
- Set X,Y to -5,+5
- Set Z to -0.01 , +0.01 for 2D
• Step 2: (((Ac)a)C) - Rotate + Translate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (y*sin(d) + c*cos(d))^2 -0.5^2 = 0
• Step 3: (((I))I) - 4x1 row of circles
(sqrt((sqrt(x^2 + a^2) - 2)^2 + b^2) - 1)^2 + y^2 -0.5^2 = 0
- A,B Translates away from center
• Step 4: (((II))) - 4x concentric circles
(sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + b^2 -0.5^2 = 0
• Step 5: (((I)I)) - 2x pair in 2x1 row of circles
(sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) - 1)^2 + b^2 -0.5^2 = 0







4D ((II)(II)) - Tiger , S1xC2 , ((maj1)(maj2)min)
---------------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2= 0
• ((II)(Ii)) - vertical stack of torii
(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0
-2.5 < a < 2.5
• ((IY)(Iy))
(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

• ((IA)(Ia)) - Rotation + Translation
(sqrt(x^2 + (y*sin(b/57.3) + a*cos(b/57.3))^2) - 3)^2 + (sqrt(z^2 + (y*cos(b/57.3) - a*sin(b/57.3))^2) - 3)^2 = 1







5D (((II)I)(II)) - Tiger Torus , S1xC2xS1 = S1x[T2*S1] , (((maj1)med1)(med2)min)
---------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) -R1a)^2 + z^2) -R2)^2 + (sqrt(w^2 + v^2) -R1b)^2 -R3^2 = 0
• (((I))(II)) - 4 torii in 1x1x4 column
(Sqrt((Sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + z^2) -2)^2 -0.5^2 = 0
• (((II))(I)) - 4 torii in 2 concentric along 1x1x2 column
(Sqrt((Sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + (Sqrt(z^2 + b^2) - 2)^2 - 0.5^2 = 0
• (((I)I)(I)) - 4 torii in 2x1x2 vertical square
(Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + y^2) -1.2)^2 + (Sqrt(z^2 + b^2) -1.5)^2 -0.4^2 = 0
• (((IY)z)(Zy)) Rotation
(Sqrt((Sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (z*cos(b))^2) - 1.2)^2 + (Sqrt((z*sin(b))^2 + (y*cos(a))^2) - 1.5)^2 - 0.5^2 = 0
(Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + b^2) -1.2)^2 + (Sqrt(y^2 + z^2) - 1.5)^2 - 0.5^2 = 0
• (((Ic)a)(AC))
(sqrt((sqrt(x^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (y*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) -2)^2 -0.5^2 = 0
• (((Ac)a)(CI))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) -2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) -2)^2 -0.5^2 = 0
• (((I))(II)) - rotates whole cut, no trans or slice rotate
(sqrt((sqrt((x*sin(b) + y*cos(a))^2 + a^2) -2)^2 + b^2) -1)^2 + (sqrt((x*cos(b) - y*sin(a))^2 + z^2) -2)^2 -0.5^2 = 0
• (((Ia)b)(YI)) from (((I))(II))
(Sqrt((Sqrt(x^2 + (y*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*((sin(a))*(sin(b))))^2 + z^2) -2)^2 -0.5^2 = 0
-- Start [A,B] at [1.57,1.57], rotate to either A or B = 0
• (((IY)a)(Ib)) from (((II))(I))
(Sqrt((Sqrt(x^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + (y*cos(b))^2) - 2)^2 - 0.5^2 = 0
• (Sqrt((Sqrt(x^2 + ((y*sin(c))*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + ((y*cos(c))*cos(b))^2) - 2)^2 - 0.5^2 = 0
• (((Ia)c)(AC)) - Dual Translate+Rotate , a/b = slide/rotate in 4D ; c/d = slide/rotate in 5D
(sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) -6)^2 + (z*cos(d) - c*sin(d))^2) -3)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) -3)^2 = 1
• (((AC)c)(Ia)) - Dual Translate+Rotate , a/b = slide/rotate in 4D ; c/d = slide/rotate in 5D
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) -6)^2 + (y*cos(d) - c*sin(d))^2) -3)^2 + (sqrt(z^2 + (x*cos(b) - a*sin(b))^2) -3)^2 = 1
XYZbox = -12 / +12
-12 < a,c < 12
0 < b,d < 1.57







5D ((((II)I)I)I) - Tritorus , T4 , ((((maj)sec)tert)min)
--------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 - R4^2= 0
• ((((I)I))I)
(sqrt((sqrt((sqrt(x^2 + a^2) - 9)^2 + y^2 ) - 4)^2 + b^2) - 2)^2 + z^2 - 1.25^2= 0
• ((((Iy)Y)z)Z)
(sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) - 9)^2 + (y*sin(a))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 - 1.25^2= 0
• ((((Iz)Y)y)Z)
(sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 9)^2 + (y*sin(b))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0
• ((((IY)z)y)Z)
(sqrt((sqrt((sqrt(x^2 + (y*sin(b))^2) - 9)^2 + (z*cos(a))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0
• ((((XY)x)I)y)
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 8)^2 + (x*cos(a))^2 ) - 4)^2 + z^2) - 2)^2 + (y*cos(b))^2 - 1.25^2= 0
• ((((XI)I)a)b)
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + y^2) - 9)^2 + z^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2 - 1= 0
• ((((XI)Z)x)z)
(sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (z*sin(b))^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (z*cos(b))^2 - 1= 0
• ((((XI)Z)z)x)
(sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (z*sin(b))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (x*cos(a))^2 - 1= 0
• ((((XI)x)z)Z)
(sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 - 1= 0
R1 Void Rotations
• ((((XY)x)y)I)
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + z^2 - 1= 0
• ((((XY)I)x)y)
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + z^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (y*cos(b))^2 - 1= 0
• ((((XY)x)I)y)
(sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + z^2) - 2)^2 + (y*cos(b))^2 - 1= 0
• (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + y^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (x*cos(b))^2 - 1= 0
--- [A,C,a,B,c,b]
• ((((XY)Z)[a,z])[y,b]) - [x->a,b] [z->c] [y->d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(d) + (x*cos(b))*cos(d))^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (y*cos(d) - (x*cos(b))*sin(d))^2 - 1 = 0
—— good morph sequence : set a,b,c,d to 1.57; slider sequence for A=0 -> a=1.57 : [D,C,d,A,c,a] and {C,A,c,a,D,C,d,c]






5D (((II)(II))I) - Toratiger , T2xC2 , (((maj1)(maj2)med)min)
-----------------------------------------------------------------------------------
(((II)(II))I)
((II)(II))I
( (II) (II)) I
( (xy) (zw)) v
( (x+y) + (z+w)) + v
( (x+y -R1a) + (z+w -R1b) -R2) + v = Rminor
( (x+y -R1a)² + (z+w -R1b)² -R2)² + v = Rminor²
( (√(x+y) -R1a)² + (√(z+w) -R1b)² -R2)² + v = Rminor²
( √((√(x+y) -R1a)² + (√(z+w) -R1b)²) -R2)² + v = Rminor²
(√((√(x²+y²) -R1a)² + (√(z²+w²) -R1b)²) -R2)² + v² = Rminor²
(sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 + v^2 = Rminor^2
• (((II)(II))I)
(sqrt((sqrt(x^2+y^2) -6)^2 + (sqrt(z^2+w^2) -6)^2) -3)^2 + v^2 = 1
• (((II)(I))) : 1x1x2x[Rminor pair]
(sqrt((sqrt(x^2+y^2) -6)^2 + (sqrt(z^2+0^2) -6)^2) -3)^2 + 0^2 = 1
• (((I)(I))I) : 2x2x1 square
(sqrt((sqrt(x^2+0^2) -6)^2 + (sqrt(y^2+0^2) -6)^2) -3)^2 + z^2 = 1
• (((AC)(Ia))c) : Dual Translate + Rotate
(sqrt((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2)-6)^2+(sqrt(z^2+(x*cos(b)-a*sin(b))^2)-6)^2)-3)^2+(y*cos(d)-c*sin(d))^2 = 1






5D ((III)II) Spherisphere , S2xS2 , (maj)min)
-----------------------------------------------
• ((III)II)
(sqrt(x^2 + y^2 + z^2) -R1)^2 + w^2 + v^2 -R2^2 = 0
R1 = 3 / R2 = 1
(sqrt(x^2 + y^2 + z^2) -3)^2 + w^2 + v^2 -1 = 0
• ((I)II) - 2x Spheres (III) in 2x1x1 row
(sqrt(x^2 + 0^2 + 0^2) -3)^2 + w^2 + v^2 -1 = 0
• ((II)I) - 1x Torus
(sqrt(x^2 + y^2 + 0^2) -3)^2 + w^2 + 0^2 -1 = 0
• ((III)) - 2x Spheres (III) in concentric pair
(sqrt(x^2 + y^2 + z^2) -3)^2 + 0^2 + 0^2 -1 = 0
• ((IYZ)yz)
(sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) -3)^2 + (y*cos(a))^2 + (z*cos(b))^2 -1 = 0





5D (((II)I)II) - Spheriditorus , S2xT2 , (((maj)med)min)
---------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2 - R3^2 = 0
• (((I))II)
(sqrt((sqrt(x^2 + a^2) - 3.5)^2 + b^2 ) - 2)^2 + y^2 + z^2 = 1^2
(sqrt((sqrt(x^2 + (y*cos(a))^2) - 4)^2 + (z*cos(b))^2 ) - 2)^2 + (y*sin(a))^2 + (z*sin(b))^2 - 1^2 = 0





5D (((III)I)I) - Ditorisphere , T2xS2 , (((maj)med)min)
--------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2))^2 - R3^2 = 0
major - R1 / medium - R2 / minor - R3
• (((IYZ)y)z)
(sqrt((sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + (z*cos(b))^2 - 0.5^2 = 0





5D (((II)II)I) - Torispheritorus , S1xS2xS1 , (((r1)r2)r3)
-----------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) -r1)^2 + z^2 + w^2) -r2)^2 + v^2 -r3^2 = 0
r1= 4 / r2= 2 / r3= 1
(sqrt((sqrt(x^2 + y^2) -4)^2 + z^2 + w^2) -2)^2 + v^2 -1 = 0
• ((()II)I) - empty, moving out makes 1x torisphere evolution of ((IIi)I)
(sqrt((sqrt(0^2 + 0^2) -4)^2 + x^2 + y^2) -2)^2 + z^2 -1 = 0
• (((I)I)I) - 2x Tori ((II)I) in 2x1x1 row
(sqrt((sqrt(x^2 + 0^2) -4)^2 + y^2 + 0^2) -2)^2 + z^2 -1 = 0
• (((I)II)) - 4x Spheres (III) as 2x1x1 row of R1 pairs
(sqrt((sqrt(x^2 + 0^2) -4)^2 + y^2 + z^2) -2)^2 + 0^2 -1 = 0
• (((II))I) - 2x Tori ((II)I) as R1 pair
(sqrt((sqrt(x^2 + y^2) -4)^2 + 0^2 + 0^2) -2)^2 + z^2 -1 = 0
• (((II)I)) - 2X Tori ((II)I) as R2 pair
(sqrt((sqrt(x^2 + y^2) -4)^2 + z^2 + 0^2) -2)^2 + 0^2 -1 = 0
• (((IY)yz)Z)
(sqrt((sqrt(x^2 + (y*sin(a))^2) -4)^2 + (y*cos(a))^2 + (z*cos(b))^2) -2)^2 + (z*sin(b))^2 -1 = 0
• (((IA)ac)C) 
(sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) -4)^2 + (y*cos(b) - a*sin(b))^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (z*sin(d) + c*cos(d))^2 -1 = 0






5D ((III)(II)) - Cylspherintigroid , S1x[S2*S1]
-----------------------------------------------------
(√(x²+y²+z²) -R1a)² + (√(w²+v²) -R1b)² = Rminor²
(sqrt(x^2+y^2+z^2) -R1a)^2 + (sqrt(w^2+v^2) -R1b)^2 = Rminor^2
--------------------------------------
((3)(3)1) - Ring-Torus Diameter Values
(sqrt(x^2+y^2+z^2) -3)^2 + (sqrt(w^2+v^2) -3)^2 = 1
3D Midsections along ((xyz)(wv))
--------------------------------
XYZboz = -6 , +6
• ((xy)(w)) - zv=0 , 1x1x2 column of tori
(sqrt(x^2+y^2+0^2) -3)^2 + (sqrt(z^2+0^2) -3)^2 = 1
• ((x)(wv)) - yz=0 , 1x1x2 column of tori
(sqrt(x^2+0^2+0^2) -3)^2 + (sqrt(y^2+z^2) -3)^2 = 1
• ((xyz)()) - wv=0 , Void R1b , ring intercepts at [±3] are 2 places of ((III)) , [R pair] of 2 spheres
(sqrt(x^2+y^2+z^2) -3)^2 + (sqrt(0^2+0^2) -3)^2 = 1
Dual Translate + Rotate Function
---------------------------------
• ((Iac)(AC)) : dual trans+rotate
(sqrt(x^2+(y*cos(b)-a*sin(b))^2+(z*cos(d)-c*sin(d))^2)-3)^2+(sqrt((y*sin(b)+a*cos(b))^2+(z*sin(d)+c*cos(d))^2)-3)^2 = 1
-6 < a,c < 6
0 < b,d < 1.57






6D (((II)I)((II)I)) - Tiger Duotorus , S1xC2xC2  (((maj1)med1)((maj2)med2)min)
-------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3^2 = 0
• (((I)I)((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2
• (((Xz)Y)((Zx)y))
(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(c))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2
• (((Xz)Y)((Zy)x))
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(b))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(b))^2) - 2)^2 + (x*cos(c))^2) -1)^2 = 0.4^2
• (((II))((I)))
(sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2
• (((XY)z)((Zx)y))
(sqrt((sqrt((x*sin(b))^2 + (y*sin(c))^2) - 2)^2 + (z*cos(a))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2
-- A,C=1.57 / B=0.785 is (((xI))((xI))), quadruple tiger cage // A,C=0 / B=0.785 is (((x)I)((x)I)) 2x oblique tiger scan
• (((IA))((Ia))) translate A , rotate B
(sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2) -1)^2 - 0.4^2 = 0
-- B=0.785 , Adjust A for flythrough of (((OI))((OI))) oblique structure, di-duoring structural scan with R3=0.2
-- XYZ = -5/+5
-- A = -4.5~4.5
-- A = 1.22 , adj B for neat topo change of six tori
• (((A)I)((Ca)c)) translate A,C  rotate B,D
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + (z*cos(d) - c*sin(d))^2) -1)^2 = 0.4^2
-- B=0.785 ; C,D=0 , Adjust A for fantastic diagonal translate along 2x1x2x1 square of tigers (((I)I)((I)I)) !!
• (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2- 0.3^2 = 0
-- c=1.16667 , adj [B,D] through 4x 90deg turns
• (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2) -1)^2 - 0.3^2 = 0
• (Sqrt((Sqrt(x^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + (y*cos(b))^2) - 2)^2 - 0.5^2 = 0
• sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2
• (sqrt((sqrt(x^2 + (y*cos(a))^2) - 2)^2 + (y*((sin(a))*(sin(b))*(sin(c))))^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2
--- Interesting, works too. At least two out of three have to be 1.57, setting one to 0 makes morphs
• (sqrt((sqrt(x^2 + ((y*cos(a))*sin(d))^2) - 2)^2 + (y*((sin(a))*(sin(b))*(sin(c))))^2) -1)^2 + (sqrt((sqrt(z^2 + ((y*cos(b))*cos(d))^2) - 2)^2 + (y*cos(c))^2) -1)^2= 0.4^2
-- Works with interesting results!






6D ((((II)I)I)(II)) - Tiger Ditorus , S1xC2xT2 / S1x[T3*S1] / T3xC2 , ((((maj)sec)tert1)(tert2)min)
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(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3a)^2 + (sqrt(v^2 + u^2) - R3b)^2 - R4^2  = 0
• ((((I))I)(I)) - ((((X))Y)(Z))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4.5)^2 + b^2) - 2.2)^2 + y^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2  = 0
• ((((II)))(I)) - ((((XY)))(Z))
(sqrt((sqrt((sqrt(x^2 + y^2) - 4.5)^2 + a^2) - 2.2)^2 + b^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2  = 0
• ((((Xz)x)Y)(Zy))
(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4.5)^2 + (x*cos(b))^2) - 2.2)^2 + (y*sin(c))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((Xy)z)Y)(Zx))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.5)^2 + (z*cos(b))^2) - 2.2)^2 + (y*sin(a))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((Xz)y)Y)(Zx))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 4.5)^2 + (y*cos(b))^2) - 2.2)^2 + (y*sin(b))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((XY)y)z)(Zx)) same as ((((Xz)x)Y)(Zy))
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(a))^2) - 4.5)^2 + (y*cos(a))^2) - 2.2)^2 + (z*cos(b))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2  = 0
• ((((Ac)a)I)(C)) - ((((I))I)(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + y^2) - 1.75)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2  = 0
• ((((A)c)a)(CI)) - ((((I)))(II))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (y*cos(d) - c*sin(d))^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2  = 0
• ((((Ia))c)(AC)) - ((((I)))(II))
(sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 5.75)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2  = 0
• ((((I)a)c)(AC)) - ((((I)))(II))
(sqrt((sqrt((sqrt(x^2 + 0^2) - 6)^2 + (y*cos(b) - a*sin(b))^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2)- 3)^2 - 0.6^2  = 0
--- a=3 , c=0 / [b,d] go thru [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0] Very interesting topology change!!!
• ((((Ia)c))(AC)) - ((((I)))(II))
(sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 6)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2  = 0
• ((((A)a)C)(Ic)) - ((((I))I)(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2  = 0
--- a=5.75 , trans out to single tigritorus, slide b for neat rotations
• ((((Ac))C)(Ia)) - ((((I))I)(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 5.75)^2 + 0^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.75^2  = 0
• ((((AC)a))(Ic)) - ((((II)))(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2  = 0
• ((((AC)a)c)(I)) - ((((II)))(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2  = 0
• ((((Ac)I)a)(C)) - ((((I)I))(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2  = 0
• ((((Ac)I))(Ca)) - ((((I)I))(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.6^2  = 0
• ((((a))I)(AI)) - (((())I)(II))
(sqrt((sqrt((sqrt((y*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + x^2) - 2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + z^2) - 2)^2 - 0.6^2  = 0
--- set b=0.785 , adj A for 4x OBLQ tiger scan along line
• ((((a))A)(II)) - (((())I)(II))
(sqrt((sqrt((sqrt((x*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(y^2 + z^2) - 2)^2 - 0.6^2  = 0
--- set B=0.785 , adj A for 4x tiger dance along line
• ((((a)c)C)(AI)) - (((())I)(II))
(sqrt((sqrt((sqrt((y*cos(d) - c*sin(d))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2  = 0
--- Very cool exploration!!!!






6D (((II)(II))(II)) -  Double Tiger ,  T2xC3 ,  (((maj1)(maj2)med)(maj3)min)
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(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3^2 = 0
• (((I)(I))(I))
(sqrt((sqrt(x^2 + a^2) - R1)^2 + (sqrt(y^2 + b^2) - R2)^2) - R3)^2 + (sqrt(z^2 + c^2) - R4)^2 - R5^2 = 0
• (((I)(I))(I))
(sqrt((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + c^2) - 3)^2 - 0.5^2 = 0
• (((Xz)(Yx))(Zy))
(sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2.5)^2 - 0.5^2 = 0
Octatangent Cut at
a = 0.36497
b = 0.3492
c = 0.91242
•(sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 0.5^2 = 0
• (((A)(Ic))(Ca))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 3)^2 + (sqrt(y^2 + (z*cos(d) - c*sin(d))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 - 0.5^2 = 0
--- Set a=3 , rotate B,D for tiger torus morphing
• (((A)(Ca))(Ic))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 3)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 3)^2 - 0.5^2 = 0
--- XYZ = -7,+7
--- Set a=-3 , c=1.85 / [b,d] go through [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0] for interesting topology morphing





6D ((II)(II)(II)) - Tritiger , S2xC3 , ((maj1)(maj2)(maj3)min)
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(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2^2 = 0
• ((I)(I)(I)) - 2x2x2 array of 8 spheres
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 - 1^2 = 0
• ((Xy)(Yz)(Zx))
(sqrt((x*sin(c))^2 + (y*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 1^2 = 0
XYZmin/max = -5,+5
• ((Xz)(Yx)(Zy))
(sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 3)^2 - 0.8805^2 = 0
-- Special diameter values for 45x45x45 cut, dodecatangent
• ((A)(Ca)(Ic)) : Dual R+T for Triger
(sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 2.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.5)^2 - 1^2 = 0





6D ((((II)(II))I)I) - Ditoratiger , T3xC2 , ((((maj1)(maj2)sec)tert)min)
-----------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0
• Diameter Adjustment Function
(sqrt((sqrt((sqrt(x^2 + 0^2) - a)^2 + (sqrt(y^2 + 0^2) - a)^2) - b)^2 + 0^2) - c)^2 + z^2  = 1
XYZbox = -20 / +20
- a = 8
- b = 4
- c = 2
• ((((I)(I)))I)
(sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + (sqrt(y^2 + b^2) - 8)^2) - 4)^2 + c^2) - 2)^2 + z^2  = 1
• ((((Xz)(Yx))y)Z)
(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 8)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 8)^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (z*sin(a))^2  = 1
• ((((Xy)(Yz))x)Z)
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 8)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 8)^2) - 4)^2 + (x*cos(c))^2) - 2)^2 + (z*sin(b))^2  = 1





6D (((II)I)(II)I) - Spheritiger Torus , S2xC2xS1 = S2x[T2*S1] , (((maj)med1)(med2)min)
------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 + u^2 - R3^2 = 0
• (((Ii)i)(Ii)I) = (((I))(I)I) - 4x2 array of 8 spheres
(sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0
• (((II)i)(Ii)i) = (((II))(I)) - 2 conc stacked 2 high of 4 torii
(sqrt((sqrt(x^2 + y^2) -2)^2 + a^2) -1)^2 + (Sqrt(z^2 + b^2) -2)^2 + c^2 -0.75^2 = 0
• (((Ii)i)(II)i) = (((I))(II)) - vert column of 4 torii
(sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0
• (((Xz)y)(Yx)Z)
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0
• (((Xy)z)(yx)Z)
(sqrt((sqrt((x*sin(c))^2 + (y*cos(b))^2) -2)^2 + (z*cos(a))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0
• (((Xy)x)(Yz)Z) - (((I))(I)I)
(sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) -4)^2 + (x*cos(b))^2) -1.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) -4)^2 + (z*sin(c))^2 -0.75^2 = 0





6D ((((II)I)(II))I) - Toratiger Torus , T2xC2xS1 , ((((maj)sec)(maj2)tert)min)
----------------------------------------------------------------------------------------------------------------
(√((√((√(x²+y²) -R1a)² +z²) -R2)² + (√(w²+v²) -R1b)²) -R3)² +u² = Rminor²
(sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2) - R3)^2 + u^2 - R4^2 = 0
((((8)4)(4)2)1)
• ((((I))(I))I) - 4x2 array of 8 torii
(sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + b^2) - 4)^2 + (sqrt(y^2 + c^2) - 4)^2) - 2)^2 + z^2 = 1
• ((((II))(I))) - 2 cocirc by 2 conc stacked 2 high of 8 torii
(sqrt((sqrt((sqrt(x^2 + y^2) - 8)^2 + a^2) - 4)^2 + (sqrt(z^2 + b^2) - 4)^2) - 2)^2 + c^2 = 1
• ((((I))(II))) - 2 cocirc stacked 4 high of 8 torii
(sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + b^2) - 4)^2 + (sqrt(y^2 + z^2) - 4)^2) - 2)^2 + c^2 = 1
• ((((I)I)(I))) - 2 cocirc in 2x1x2 vert square of 8 torii
(sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + y^2) - 4)^2 + (sqrt(z^2 + b^2) - 4)^2) - 2)^2 + c^2 = 1
• ((((Xy)z)(Yx))Z)
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 8)^2 + (z*cos(b))^2) - 4)^2 + (sqrt((y*sin(a))^2 + (x*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(b))^2 = 1
• ((((Xz)y)(Yx))Z)
(sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 8)^2 + (y*cos(b))^2) - 4)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(a))^2 = 1
• ((((Xy)x)(Yz))Z)
(sqrt((sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) - 8)^2 + (x*cos(b))^2) - 4)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(c))^2 = 1
• ((((Ac)I)(a))C)
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 8)^2 + y^2) - 4)^2 + (sqrt(0^2 + (x*cos(b) - a*sin(b))^2) - 4)^2) - 2)^2 + (z*sin(d) + c*cos(d))^2 = 1
-- Very good exploration of ((((I)I)(I))I) cut, though Z rotations make major-->minor concentric morph
• ((((A)C)(ac))I)
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 8)^2 + (y*sin(d) + c*cos(d))^2) - 4)^2 + (sqrt((x*cos(b) - a*sin(b))^2 + (y*cos(d) - c*sin(d))^2) - 4)^2) -2)^2 + z^2 = 1
-- A= -1.29 , C= 0.9 : Adjust B,D angles to all four combinations, very cool stuff!
• ((((A)c)(Ia))C)
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-8)^2+(z*cos(d)-c*sin(d))^2)-4)^2+(sqrt(y^2+(x*cos(b)-a*sin(b))^2)-4)^2)-2)^2+(z*sin(d)+c*cos(d))^2 = 1
• ((((Ac))(Ia))C) - 4x2 array of 8 torii
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2 + (z*cos(d)-c*sin(d))^2) - 8)^2 + 0^2) - 4)^2 + (sqrt(y^2 + (x*cos(b)-a*sin(b))^2) - 4)^2) - 2)^2 + (z*sin(d)+c*cos(d))^2 = 1






7D ((((II)(II))I)(II)) - Tigritiger , T3xC3 = S1xC2xS1xC2 = S1x[(T2xC2)*S1] , ((((maj1)(maj2)sec)tert)(maj3)min)
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(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2 = Rminor^2
• ((((I)(I)))(I)) : 2x2x2x[R1 pair] array of 16 tori
(sqrt((sqrt((sqrt(x^2+0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1
———— XYZbox = -17/+17
• ((((A)(Ic))a)(C))
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -7.5)^2 + (sqrt(y^2+(z*cos(d)-c*sin(d))^2) -7.5)^2) -3.5)^2 +(x*cos(b)-a*sin(b))^2) -1.5)^2 + (sqrt((z*sin(d)+c*cos(d))^2+0^2) -3.5)^2 = 1
——— -15 < a,c < 15
——— 0 < b,d < 1.5707
• ((((Xc)(I))a)(Cb))
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+(z*cos(d) - c*sin(d))^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(a))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(b))^2) -3.5)^2 = 1
— -15 < c < 15
— 0 < a,b,d < 1.5707
• ((((X)(Ia))c)(Cb))
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+0^2) -7.5)^2 + (sqrt(y^2+(x*cos(a))^2) -7.5)^2) -3.5)^2 +(z*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(a))^2) -3.5)^2 = 1
• ((((A)(Ca))c)(I))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +(y*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1
——— -17 < a,c < 17
——— 0 < b,d < 1.5707
• ((((A)(Ca)))(Ic)) ** good function **
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1
——— -17 < a,c < 17
——— 0 < b,d < 1.5707
• ((((A)(C))a)(Ic))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(b) - a*sin(b))^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1
——- a=7.5,c=7.5±3.5 for 1x1x2 column of tori in cut (((()())I)(II))





7D (((II)I)((II)I)I) - Spheritiger Duotorus , S2xC2xC2 = S2x[T2*T2] , (((maj1)med1)((maj2)med2)min)
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(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 + t^2 - R3^2 = 0
• (((I))((I))I)
(sqrt((sqrt(x^2 + a^2) - 2.35)^2 + b^2) -1.15)^2 + (sqrt((sqrt(y^2 + c^2) - 2.6)^2 + d^2) -1.25)^2 + z^2 - 0.5^2 = 0
• (((Xz)y)((Yx)d)Z)
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2.35)^2 + (y*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.6)^2 + d^2) -1.25)^2 + (z*sin(a))^2 - 0.5^2 = 0
• (((Xy)z)((Yc)x)Z)
(sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 2.35)^2 + (z*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(a))^2 + c^2) - 2.6)^2 + (x*cos(d))^2) -1.25)^2 + (z*sin(b))^2 - 0.5^2 = 0
• (((Xa)x)((Yz)y)Z)
(sqrt((sqrt((x*sin(b))^2 + a^2) - 2.35)^2 + (x*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(d))^2 + (z*cos(c))^2) - 2.6)^2 + (y*cos(d))^2) -1.25)^2 + (z*sin(c))^2 - 0.5^2 = 0





7D ((((II)I)(II)I)I) - Torispheric Tigritorus , S1xS2xC2xS1 = S1xS2x[T2*S1] , ((((maj1)sec)(maj2)tert)min)
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(sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0
• Diameter Adjustment Function
(sqrt((sqrt((sqrt(x^2 + 0^2) - a)^2 + 0^2) - b)^2 + (sqrt(y^2 + 0^2) - b)^2 + 0^2) - c)^2 + z^2 = 1
- XYZbox = -22 / +22
- a = 10
- b = 5
- c = 3
• ((((I))(I))I) - 8x torii ((II)I) stacked in 4x2 major rectangle array
(sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + c^2) - 5)^2 + d^2) - 3)^2 + z^2 = 1
• ((((I))(I)I)) - 16x spheres (III) as concentric pair stacked in 4x2 rectangle array
(sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + c^2) - 5)^2 + z^2) - 3)^2 + d^2 = 1
• ((((I))(II))) - 8x torii ((II)I) as concentric minor pair stacked in 1x1x4 minor column
(sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + z^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1
• ((((I)I)(I))) - 8x torii ((II)I) as concentric minor pair stacked in 2x1x2 maj/min square array
(sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + y^2) - 5)^2 + (sqrt(z^2 + b^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1
• ((((II))(I))) - 8x torii ((II)I) as concentric major/minor pairs stacked in 1x1x2 minor column
(sqrt((sqrt((sqrt(x^2 + y^2) - 10)^2 + a^2) - 5)^2 + (sqrt(z^2 + b^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1
• ((((Xy)z)(Yc)x)Z) - triple rotate a,b,d with sliding c
(sqrt((sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 10)^2 + (z*cos(b))^2) - 5)^2 + (sqrt((y*sin(a))^2 + c^2) - 5)^2 + (x*cos(d))^2) - 3)^2 + (z*sin(b))^2 = 1
• ((((Xz)x)(Yi)y)Z) - triple rotate a,b,d with sliding c
(sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 10)^2 + (x*cos(b))^2) - 5)^2 + (sqrt((y*sin(d))^2 + c^2) - 5)^2 + (y*cos(d))^2) - 3)^2 + (z*sin(a))^2 = 1
• ((((Xz)y)(Yx)Z)i) - triple rotate a,b,c with sliding d
(sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 10)^2 + (y*cos(b))^2) - 5)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 5)^2 + (z*sin(a))^2) - 3)^2 + d^2 = 1
• ((((Ai)c)(Ca)i)I) - dual translate + rotate
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 10)^2 + (y*cos(d) - c*sin(d))^2) - 5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + z^2 = 1
• ((((Az)y)(Ya)i)Z) - Trans+Rotate A,a / Dual rotate c,d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d))^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + (z*sin(d))^2 = 1
• ((((Ai)y)(Ya)z)Z) - , Trans+Rotate A,a / Dual rotate c,d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + (z*cos(d))^2) - 3)^2 + (z*sin(d))^2 = 1
• ((((Ad)y)(Ya)i)I) - , Trans+Rotate A,a / rotate c / translate d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + d^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + z^2 = 1







7D ((((II)I)((II)I))I) - Toratiger Duotorus , T2xC2xC2 = T2x[T2*T2] , ((((maj1)sec1)((maj2)sec2)tert)min)
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(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2) -R3)^2 +t^2 -R4^2 = 0
• ((((I))((I)))I) - 16 tori in 4x4x1 maj array - Diameter Adjustment Equation
(sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt((sqrt(y^2+0^2) -a)^2 +0^2) -b)^2) -c)^2 +z^2 -d^2 = 0
XYZbox = -22 / +22
- a = 10
- b = 5
- c = 3
- d = 1
• ((((I))((I)))I) - 16 tori in 4x4x1 maj array
(sqrt((sqrt((sqrt(x^2+0^2) -10)^2 +0^2) -5)^2 + (sqrt((sqrt(y^2+0^2) -10)^2 +0^2) -5)^2) -3)^2 +z^2 = 1
(sqrt((sqrt((sqrt(x^2+a^2) -10)^2 +b^2) -5)^2 + (sqrt((sqrt(y^2+c^2) -10)^2 +d^2) -5)^2) -3)^2 +z^2 = 1
--- XYZ = -12,+12
• ((((Xy)z)((Yc)x))Z) - ((((I))((I)))I)
(sqrt((sqrt((sqrt((x*sin(a))^2+(y*cos(b))^2) -10)^2 +(z*cos(c))^2) -5)^2 + (sqrt((sqrt((y*sin(b))^2+0^2) -10)^2 + (x*cos(a))^2) -5)^2) -3)^2 +(z*sin(c))^2 = 1
• ((((Ac))((C)a))I) - ((((I))((I)))I) --> Explores ((((II)I)((I)))I) , a 1x1x1x4x1 tertiary column of 4x tritoruses
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(d) - c*sin(d))^2) -10)^2) -5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -10)^2 +(x*cos(b) - a*sin(b))^2) -5)^2) -3)^2 +z^2 = 1
• ((((A)a)((C)c))I) - ((((I))((I)))I)
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 +(x*cos(b) - a*sin(b))^2) -5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -10)^2 +(y*cos(d) - c*sin(d))^2) -5)^2) -3)^2 +z^2 = 1






7D ((((II)I)I)((II)I)) - Tiger Duotoric Torus , S1xC2xC2xS1 , ((((R1)R2a)R3a)((R2b)R3b)R4)
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• ((((II)I)I)((II)I))
(sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + w^2) - R3a)^2 + (sqrt((sqrt(v^2 + u^2) - R2b)^2 + t^2) - R3b)^2 - R4^2 = 0
(sqrt((sqrt((sqrt(x^2 + y^2) - 4.25)^2 + z^2) - 2)^2 + w^2) - 1)^2 + (sqrt((sqrt(v^2 + u^2) - 2.5)^2 + t^2) - 1.25)^2 - 0.4^2 = 0
--- XYZ= -9,+9
• ((((I)))((I)I)) - 16x Tori ((II)I) in 2x1x8 vert rectangle array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + 0^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.4^2 = 0
• ((((I)))((II))) - 16x Tori ((II)I) as R1 conc pair in 1x1x8 column
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + z^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0
• ((((I))I)((I))) - 16x Tori ((II)I) in 4x1x4 vert square array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0
• ((((I)I))((I))) - 16x Tori ((II)I) as R1 conc pair in 2x1x4 vert rectangle array
(sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + y^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0
• ((((II)))((I))) - 16x Tori ((II)I) as R1 conc quartet in 1x1x4 column
(sqrt((sqrt((sqrt(x^2 + y^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0
• ((((IA)a))((C)c))
(sqrt((sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 4)^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.5)^2 + (z*cos(d) - c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0
• ((((AC))a)((I)c))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 4)^2 + 0^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + (y*cos(d) - c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0
• ((((XY)z)a)((Zb)y)) - new multi-position rotate with [X -> a,b][Y->c][Z->d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(c))^2) - 4.25)^2 + (z*cos(d))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(a))^2) - 2.5)^2 + (y*cos(c))^2) - 1.25)^2 - 0.4^2 = 0
--- A=1.0635 , B=1.57 , [C,D] => 4x Rotation Cycle , or [C,D] double rotate 0->1.57
--- A=1.0635 , [B,C,D] => 6x Rotation Cycle, has two different cycles, very amazing!
--- A,B,C,D = 1.57 => Rotation sequence to 0 and back to 1.57 : [B,C,B,D,A,C,D,A]
• ((((XY)y)b)((Zx)a)) [Z->a,b][X->c][Y->d]
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(d))^2) - 4.25)^2 + (y*cos(d))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*((sin(a))*(sin(b))))^2 + (x*cos(c))^2) -2.5)^2 + (z*cos(a))^2) - 1.25)^2 - 0.4^2 = 0
--- Very cool with nice alternating midcut sequences
• ((((Xa)z)Y)((Zx)b)) [Y->a,b][X->c][Z->d]
(sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.25)^2 + (z*cos(d))^2) - 2)^2 + (y*((sin(a))*(sin(b))))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(c))^2) -2.5)^2 + (y*cos(b))^2) - 1.25)^2 - 0.4^2 = 0
--- A,B,C,D=1.57 => Rotation Loop Slider Sequence: [A,C,D,A,B,C,D,B] and [A,C,D,C,B,A,D,B]
--- A=0 , B=1.57 => 4x Rotation Loop Sequence [C,D,C,D]=[0->1.57] <ANIMATED>
• ((((Xy)a)b)((YZ)z)) - [X->a,b][Y->c][Z->d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*cos(c))^2) - 4.25)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(c))^2 + (z*sin(d))^2) -2.5)^2 + (z*cos(d))^2) - 1.25)^2 - 0.4^2 = 0
• ((((Ic)a)b)((C)Z)) - [Z->a,b] , Y->[C-translate+D-rotate] -----> *****AWESOME EXPLORE FUNCTION , DOES A LOT!!!*****
(sqrt((sqrt((sqrt(x^2 + (y*cos(d) - c*sin(d))^2) - 4.25)^2 + (z*cos(a))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + 0^2) - 2.5)^2 + (z*((sin(a))*(sin(b))))^2) - 1.25)^2 - 0.4^2 = 0
——- XYZ = -9 , +9
——- a,b,d = 0 ~ 1.57 / c = -10 ~ +10
——- [a,b] are multi-position rotate : [1.57,1.57] = ((((I)))((I)I)) / [1.57,0] = ((((I))I)((I))) / [0,1.57] = ((((I)I))((I))) / [0,0] = invalid/no solution
——- [c,d] are translate+rotate
--- 6-Step D rotation, animate C from -9 -> +9 for scanning the column of 4x tritoruses ((((II)I)I)((I)))
--- [a,b] Gives all three cuts of (((II)I)(II)) intercept while in square array ((((I)I)I)((I)I))
--- c=-1.4 , Adjust a,b,d for awesome morphs of translated empty rotations, has triple rotate parameter on 5D tritorus intercept
• ((((C)a)b)((Ic)Z))  - X->[c-trans,d-rot] / Z-start->[a-end1 , b-end2]
(sqrt((sqrt((sqrt((x*sin(d) + c*cos(d))^2 + 0^2) - 4.25)^2 + (z*cos(a))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt(y^2 + (x*cos(d) - c*sin(d))^2) - 2.5)^2 +(z*((sin(a))*(sin(b))))^2) - 1.25)^2 - 0.4^2 = 0
--- Nice mirror equation for alternate explore scans. Does a few more step rotation empty scans
• ((((A)y)z)((Ya)Z)) : X->[a,b] , Y->c , Z->d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 4.0)^2 + (y*cos(c))^2) - 2)^2 + (z*cos(d))^2) - 1)^2 + (sqrt((sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + (z*sin(d))^2) - 1.25)^2 - 0.4^2 = 0
• ((((Xc))I)((Ca)b)) : X->[a,b] , Z->[c+d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) - 4)^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) - 2.5)^2 +(x*cos(b))^2) - 1.25)^2 - 0.4^2 = 0
- more empty scans not seen before, many empty rotates
• ((((Xc)a)b)((CI))) : X->[a,b] , Y->[c+d]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*cos(d) - c*sin(d))^2) - 4)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0
• ((((A)c))((Ca)I)) : Dual Trans+Rotate
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 4.0)^2 + (y*cos(d)-c*sin(d))^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt((y*sin(d)+c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.4^2 = 0
• ((((IY)a)b)((Ic)d)) : Y -> a,b,c,d ; Five Position Rotate
(sqrt((sqrt((sqrt(x^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2) - 4.25)^2 + (y*cos(a))^2) - 2)^2 + (y*cos(b))^2) - 1)^2 + (sqrt((sqrt(z^2 + (y*cos(c))^2) - 2.5)^2 + (y*cos(d))^2) - 1.25)^2 - 0.4^2 = 0






7D ((((II)I)(II))(II)) - Double Tiger R1A-Torus , T3xC3 / T2xC3xS1
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Diameter Size Hierarchy : ((((R1)R2a)(R2b)R3a)(R3b)R4)
(sqrt((sqrt((sqrt(x^2 + y^2) -R1)^2 + z^2) -R2a)^2 + (sqrt(w^2 + v^2) -R2b)^2) -R3a)^2 + (sqrt(u^2 + t^2) -R3b)^2 -R4^2 = 0
R1 = 4 / R2 = 2 / R3 = 1.2 / R4 = 0.5
(sqrt((sqrt((sqrt(x^2 + y^2) -4)^2 + z^2) -2)^2 + (sqrt(w^2 + v^2) -2)^2) -1.2)^2 + (sqrt(u^2 + t^2) -1.2)^2 -0.5^2 = 0
• ((((a)b)(b)c)(c)d) - Diamter Adjustment Function
(sqrt((sqrt((sqrt(x^2 + 0^2) -a)^2 + 0^2) -b)^2 + (sqrt(y^2 + 0^2) -a)^2) -c)^2 + (sqrt(z^2 + 0^2) -a)^2 -d^2 = 0
• ((((I))(I))(I))
(sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + 0^2) -2)^2 + (sqrt(y^2 + 0^2) -2)^2) -1.2)^2 + (sqrt(z^2 + 0^2) -1.2)^2 -0.5^2 = 0
— XYZ = -8 , +8
(sqrt((sqrt((sqrt(x^2 + a^2) -4)^2 + b^2) -2)^2 + (sqrt(y^2 + c^2) -2)^2) -1.2)^2 + (sqrt(z^2 + d^2) -1.2)^2 -0.5^2 = 0
• ((((X)a)(Ib))(Ic)) : X,a,b,c : Four Position Rotate Function from R1a Void
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))))^2 + 0^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(y^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (x*cos(c))^2) -1.2)^2 -0.5^2 = 0
• ((((Ac)d)(I))(Za)): TransRot[A-{b}-a]  MPosRot[Z,c,d]
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt(y^2 + 0^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -1.2)^2 -0.5^2 = 0
--- Has interesting empty cut scans, new morphs not seen before. Strong correlation to specific void.
• ((((Ac))(Ya))(Id)) : X{A,a}  Y,c,d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(c))^2) -4)^2 + 0^2) -2)^2 + (sqrt((y*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d))^2) -1.2)^2 -0.5^2 = 0
--- Second type like above ((((Ac)d)(I))(Za)) , scans past 4x tiger toruses as column of 4 tori
• ((((XD)a)(Ib))(Ic)) : [X,a,b,c]  [D-trans]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))))^2 + d^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(y^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (x*cos(c))^2) -1.2)^2 -0.5^2 = 0
• ((((X)a)(Cb))(Ic)) : [X,a,b]  Y[C,c]
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + 0^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -1.2)^2 -0.5^2 = 0
• ((((I)d)(Ac))(Za)) : Y[A,a]  [Z,c,d]
(sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (y*cos(b) - a*sin(b))^2) -1.2)^2 -0.5^2 = 0
• ((((Id)a)(Ac))(Z)) : Y[A,a]  [Z,c,d]
(sqrt((sqrt((sqrt(x^2 + (z*cos(d))^2) -4)^2 + (y*cos(b) - a*sin(b))^2) -2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + 0^2) -1.2)^2 -0.5^2 = 0
• ((((I)A)(ac))(C)) : Y[A,a]  Z[C,c]
(sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + (y*sin(b) + a*cos(b))^2) -2)^2 + (sqrt((y*cos(b) - a*sin(b))^2 + (z*cos(d) - c*sin(d))^2) -2)^2) -1.2)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) -1.2)^2 -0.5^2 = 0
--- Set C=1 , Scan B [0~1.57] does 1x Toratiger Torus morph of ((((I)y)(YI)))
• ((((AC))(a))(Ic)) : X[A,a]  Y[C,c]
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) -4)^2 + 0^2) -2)^2 + (sqrt((x*cos(b) - a*sin(b))^2 + 0^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -1.2)^2 -0.5^2 = 0
--- Cool Rotation scans, Translate Step-Scans
• ((((XC)a)(Ic))(b)) : X,a,b  C,c : MPR + T/R
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(d) + c*cos(d))^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -2)^2) -1.2)^2 + (sqrt((x*cos(b))^2 + 0^2) -1.2)^2 -0.5^2 = 0
--- Still needs more surveying, unsure of scan morphs
• ((((Ac)d)(Ia))(Z)) : TransRot[A-{b}-a]  MPosRot[Z,c,d]
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt(y^2 + (x*cos(b) - a*sin(b))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + 0^2) -1.2)^2 -0.5^2 = 0
--- interesting rotation on parameter b
• ((((A)c)(Id))(Za)) ***** Very Good Explore Function *****
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.8)^2 + (z*cos(c))^2) -2.4)^2 + (sqrt(y^2 + (z*cos(d))^2) -3.25)^2) -1.6)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -1.6)^2 -0.58^2 = 0
——- XYZ box to -10,+10
--- Good linking with Z rotations and A scans
• ((((Ic)d)(A))(Za)) : Y[A,a]  [Z,c,d]  ***** Very Good Explore Function *****
(sqrt((sqrt((sqrt(x^2 + (z*cos(c))^2) -4.8)^2 + (z*cos(d))^2) -2.4)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + 0^2) -3.25)^2) -1.6)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (y*cos(b) - a*sin(b))^2) -1.6)^2 -0.58^2 = 0
--- A,B,C,D = [2.4,0,0,1.57] : rotations to 6D ring intercept of ((((II)I)I)(II)) in empty cut ((((II))())(I))
--- A,B,C,D = [1.6,1.57,0,1.57] : rotations to 6D ring intercept ((((II)I)(II))I) in empty cut ((((II))(I))())







7D (((II)(II))((II)I)) - Double Tiger 1C-Torus , T2xC3xS1 , S1xC2xS1xC2
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• (((II)(II))((II)I)) - (((R1A)(R1B)R2)((R1C)R3)R4)
(sqrt((sqrt(x^2 + y^2) -R1a)^2 + (sqrt(z^2 + w^2) -R1b)^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1c)^2 + t^2) -R2)^2 -R4^2 = 0
R1 = 4.25 / R2 = 2.125 / R3 = 2.55 / R4 = 0.75
XYZ = -10 , +10
• (((a)(a)c)((a)b)d) - Diameter Adjustment Function
(sqrt((sqrt(x^2 + 0^2) -a)^2 + (sqrt(y^2 + 0^2) -a)^2) -c)^2 + (sqrt((sqrt(z^2 + 0^2) -a)^2 + 0^2) -b)^2 -d^2 = 0
--- D,A,B,C seq for neat build up by diameters from 0
• (((I)(I))((I))) - 16x Tori ((II)I) in 2x2x4 vertical tower
(sqrt((sqrt(x^2 + 0^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + 0^2) -4.25)^2 + 0^2) -2.125)^2 -0.75^2 = 0
• (((Ia)(Ib))((Ic)d)) - 4-Axis Translate out of 3D
(sqrt((sqrt(x^2 + a^2) -4.25)^2 + (sqrt(y^2 + b^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + c^2) -4.25)^2 + d^2) -2.125)^2 -0.75^2 = 0
• (((A)(Ca))((I)c)) - X[A,a]   Y[C,c]
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.25)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + 0^2) -4.25)^2 + (y*cos(d) - c*sin(d))^2) -2.125)^2 -0.75^2 = 0
• (((Ac)(I))((Za)d)) : X[A,a]  [Z,c,d]
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2 + (z*cos(d))^2) -2.125)^2 -0.75^2 = 0
--- Very good explore function, notable scans of empties, plus a 4x2 of 8 tigers ETE scan
• (((A)(Ic))((Za)d)) : X[A,a]  [Z,c,d]
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.25)^2 + (sqrt(y^2 + (z*cos(c))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2 + (z*cos(d))^2) -2.125)^2 -0.75^2 = 0
--- Nice alt function to (((Ac)(I))((Za)d)) , has a few new empty scans
• (((Xc)(I))((Ca)b))
(sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) -4.25)^2 + (x*cos(b))^2) -2.125)^2 -0.75^2 = 0
--- Use A,B to change scan C of step rotation D
--- A,B = [1.57,0] / Step Rotate D 0 to 1.57 with scanning C -12 to +12 : Very nice 2x2x2 cube array of 8 tigers scan, ETE rotation
• (((Xc)(Ib))((Ca)))
(sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) -4.25)^2 + (sqrt(y^2 + (x*cos(b))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) -4.25)^2 + 0^2) -2.125)^2 -0.75^2 = 0
--- Use A,B to change scan C of step rotation D
--- Has some neat 4-cycle sequences






7D (((((II)(II))I)I)I) :  Tritoratiger : T4xC2 , (((((R1a)(R1b)R2)R3)R4)minor)
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(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2
• Diameter Adjustment Equation
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 +z^2 = 1
• (((((I)(I))))I) - 2x2x[R1quartet] array of 16 tori
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 + (sqrt(y^2+0^2) -15)^2) -7)^2 +0^2) -3.2)^2 +0^2) -1.6)^2 +z^2 = 1
—— XYZbox = -35 / +35
—— 55 cubes
• (((((A)(Ia))c)d)Z) : X -> A,a  /  Z -> c,d
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -15)^2 + (sqrt(y^2+(x*cos(b) - a*sin(b))^2) -15)^2) -7)^2 +(z*cos(c))^2) -3.2)^2 +(z*cos(d))^2) -1.6)^2 +(z*((sin(c))*(sin(d))))^2 = 1
—— -35 < a < 35
—— 0 < b,c,d < 1.5707
• (((((Xc)(Ix))b)a)Z) : X -> x  /  Z -> a,b,c
(sqrt((sqrt((sqrt((sqrt((x*sin(d))^2+(z*cos(c))^2) -15)^2 + (sqrt(y^2+(x*cos(d))^2) -15)^2) -7)^2 +(z*cos(b))^2) -3.2)^2 +(z*cos(a))^2) -1.6)^2 +(z*((sin(a))*(sin(b))*(sin(c))))^2 = 1
—— 0 < a,b,c,d < 1.5707





7D (((((II)I)(II))I)I) : Ditoratiger Torus, T3xC2xS1
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(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2
• (((((I))(I)))I) : 16x Tori in 4x2x[R1 quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +z^2 = 1
• (((((I))(I))I)) : 16x Tori in 4x2x[R1 pair]x[Rm pair]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +z^2) -1.75)^2 +0^2 = 1
• (((((I))(II)))) : 16x Tori in 1x1x4x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+z^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1
• (((((I)I)(I)))) : 16x Tori in 2x1x2x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +y^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1
• (((((II))(I)))) : 16x Tori in 1x1x2x[R1 pair]x[Rm quartet]
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -14)^2 +0^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1
• Diameter Adjustment Equation of (((((I))(I)))I)
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt(y^2+0^2) -b)^2) -c)^2 +0^2) -d)^2 +z^2 = 1
a=14 ; b=7 ; c=3.5 ; d=1.75
--- XYZbox = -30,+30





8D ((((II)I)I)(((II)I)I)) : Duoditorus Tiger , S1xC2xC2xC2
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(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 +w^2) -R3a)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -R1b)^2 +t^2) -R2b)^2 +s^2) -R3b)^2 = Rminor^2

(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +z^2) -5)^2 +w^2) -2.5)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -10)^2 +t^2) -5)^2 +s^2) -2.5)^2 = 0.5
• ((((II)))(((I))))
(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+0^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
— XYZbox = -23 / +23
-- 40 cubes
• ((((IA)))(((Ia)))) : y -> A,a
(sqrt((sqrt((sqrt(x^2+(y*sin(b) + a*cos(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
• ((((AY)c)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*((sin(c))*(sin(d))))^2) -10)^2 +(y*cos(c))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 1
• ((((Ac)Y)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(c))^2) -10)^2 +(y*((sin(c))*(sin(d))))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 1
• ((((IY)a)b)(((Ic)d)))
(sqrt((sqrt((sqrt(x^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2) -10)^2 +(y*cos(a))^2) -5)^2 +(y*cos(b))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(c))^2) -10)^2 +(y*cos(d))^2) -5)^2 +0^2) -2.5)^2 = 1





8D ((II)(II)(II)(II)) - Tetratiger , S3xC4 , ((maj1)(maj2)(maj3)(maj4)min)
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(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 + (sqrt(t^2 + s^2) - R1d)^2- R2^2 = 0
• ((I)(I)(I)(I)) - 2x2x2x2 array of 16 glomes
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 + (sqrt(d^2 + 0^2) - 2.5)^2 - 1^2 = 0
• ((Ia)(Ib)(I)(Cd))
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + 0^2) - 2.5)^2 + (sqrt(c^2 + d^2) - 2.5)^2 - 1^2 = 0
-- Adjusting C translates away from center of 2x2x2x2 array of glomes, passing by either +W or -W 2x2x2 cube array
-- Adjusting D controls merging of +W,-W 2x2x2 arrays in 4-space
• ((Xc)(Yz)(Zx)(Cy))
(sqrt((x*sin(0))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (y*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Set C=3.5 for only visible morphings.
• ((Xc)(Yx)(Zy)(Cz))
(sqrt((x*sin(b))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(0))^2 + (x*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(d))^2 + (y*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (z*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Adjusting B scans along diagonal 2x2x2 array, max illum at 0.785/45deg
-- Adjusting C translates away fr center of 2x2x2x2 array, max illum at +2.5,-2.5 in 4-space

• ((I)(A)(Ca)(c)) - Dual Translate+Rotate of tesseract array of hyperspheres; Explores a 2x2x2x2 array of 16 S^3’s
(sqrt(x^2)-3)^2+(sqrt((y*sin(b)+a*cos(b))^2)-3)^2+(sqrt((z*sin(d)+c*cos(d))^2+(y*cos(b)-a*sin(b))^2)-3)^2+(sqrt((z*cos(d)-c*sin(d))^2)-3)^2 = 1






8D ((((II)(II))I)((II)I)) : [Toratiger-Torus] Tiger , S1xC2xS1xS1xC2 , T2xC2xC3 , S1xC2xS1xC3
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(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R3)^2 +v^2) -R4)^2 + (sqrt((sqrt(u^2+t^2) -R1c)^2 +s^2) -R2)^2 = Rm^2
• Trace Array Diameter Adjustment Equation
(sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) -b)^2 +0^2) -(b/2))^2 = 1
• ((((I)(I)))((I))) : 2x2x4x[R1 pair] array of 32 toruses
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— XYZbox = -22 / +22
• ((((I)(I)))((I))) : standard explore function slate
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
• ((((A)(Ca)))((Ic))) : X -> A,a / Y -> C,c
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+(y*cos(d) - c*sin(d))^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((Ic)(A))a)((C))) : Y -> A,a / Z -> C,c
(sqrt((sqrt((sqrt(x^2+(z*cos(d) - c*sin(d))^2) -9)^2 + (sqrt((y*sin(b) + a*cos(b))^2+0^2) -9)^2) -4.5)^2 +(y*cos(b) - a*sin(b))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((A)(I))c)((C)a)) : X -> A,a / Z -> C,c
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +(z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +(x*cos(b) - a*sin(b))^2) -3.25)^2 = 1






8D ((((II)I)(II))((II)I)) - Tigritorus Duotoric Torus , T2xC2xC3 = S1x[(S1xC2xS1)*T2] , ((((maj1)sec1)(sec2)tert)((maj2)sec3)min)
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((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3)^2 + (sqrt((sqrt(u^2 + t^2) - R1b)^2 + s^2) - R2c)^2 - R4^2 = 0
• ((((I))(I))((I)))
((sqrt((sqrt(x^2 + a^2) - 5)^2 + 0^2) - 2.25)^2 + (sqrt(y^2 + b^2) - 2.25)^2 - 1.8)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.75^2 = 0






9D (((II)I)((II)I)((II)I)) - Triger Triotorus , S2xC3xC3 = S2x[T2*T2*T2] , (((maj1)med1)((maj2)med2)((maj3)med3)min)
------------------------------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2c)^2 = R3^2
• (((I))((I))((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(y^2 + b^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.5^2 = 0
• (((A))((Ca)c)((I))) - Dual Trans+Rotate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + 0^2) -2)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((sqrt(z^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1
• (((A)c)((Ia))((c))) - Dual Trans+Rotate
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt(y^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + 0^2) - 2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1






9D (((((II)I)(II))I)((II)I)) - [ToratigerTorus-Torus] Tiger , T3xC2xC3 = S1xC2xT2xC2xS1
---------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2b)^2 = Rminor^2
• Diameter Adjustment Equation for Trace Array
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) - b)^2 +0^2) - (b/2))^2 = 1
--- a=15
--- b=7
--- c=4
--- d=1.75
• (((((I))(I)))((I))) : 4x2x4x[R1 pair] of 64 Toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1
—— XYZbox = -32 / +32
—— 55 cubes
• (((((I))(I)))((I))) : Exploratory Function Template
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1

• (((((A)c)(Ca)))((I))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2+(y*cos(d)-c*sin(d))^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2)-3.5)^2 = 1
--- -30 < a < 30 ; -20 < c < 20 ; 0 < b,d < 1.57

• (((((A))(C))c)((Ia))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2)-3.5)^2 = 1
((((())(I)))((II))) : [±R1A±R2] Intercepts are 4 places of ((((I)))((II))) - 1x1x8x[R1 pair] of 16 tori

• (((((A))(C))a)((I)c)) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1

• (((((A))(Ca)))((Ic))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(y*cos(d)-c*sin(d))^2)-7)^2)-3.5)^2 = 1

• (((((A))(C)))((Ia)c)) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1

• (((((A))(C))c)((a)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt((x*cos(b)-a*sin(b))^2)-7)^2+z^2)-3.5)^2 = 1

• (((((A))(C))a)((c)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt((y*cos(d)-c*sin(d))^2)-7)^2+z^2)-3.5)^2 = 1






9D ((((((II)I)(II))I)I)(II)) : [DitoratigerTorus-Circle] Tiger , S1xC2xT2xC2xS1
----------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2) -R5)^2 + (sqrt(s^2+r^2) -R1c)^2 = Rminor^2
• ((((((I))(I))))(I)) Diameter Adjustment Function
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 + (sqrt(z^2+0^2) -(a/5))^2 = 1
--- a=27 , b=7.1 , c=3.2 , d=1.6
--- XYbox = -55 / +55 , Zbox = -25 / +25






9D (((((II)I)(II))(II))(II)) : Triple Tiger 1A-Torus , T3xC4xS1
------------------------------------------------------------------
(II) - S1
((II)I) - T2
(((II)I)I) - T3
((((II)I)I)I) - T4
((((II)(II))I)I) - T3xC2
((((II)(II))(II))I) - T3xC3
((((II)(II))(II))(II)) - T3xC4
(((((II)I)(II))(II))(II)) - T3xC4xS1
--------------------------------------
(((((II)I)(II))(II))(II))
((((II)I)(II))(II))(II)
( ( ( (II) I) (II)) (II)) (II)
( ( ( (xy) z) (wv)) (ut)) (sr)
( ( ( (x+y) +z) + (w+v)) + (u+t)) + (s+r)
( ( ( (x+y -R1a) +z -R2) + (w+v -R1b) -R3) + (u+t -R1c) -R4) + (s+r -R1d) = Rminor
( ( ( (x+y -R1a)² +z -R2)² + (w+v -R1b)² -R3)² + (u+t -R1c)² -R4)² + (s+r -R1d)² = Rminor²
( ( ( (√(x+y) -R1a)² +z -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( ( ( √((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( ( √((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor²
( √((√((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)²) -R4)² + (√(s+r) -R1d)² = Rminor²
(√((√((√((√(x²+y²) -R1a)² +z²) -R2)² + (√(w²+v²) -R1b)²) -R3)² + (√(u²+t²) -R1c)²) -R4)² + (√(s²+r²) -R1d)² = Rminor²
----------------------------------------------------------------------------------------------------------------------
• (√((√((√((√(x²+y²)-R1a)²+z²)-R2)²+(√(w²+v²)-R1b)²)-R3)²+(√(u²+t²)-R1c)²)-R4)²+(√(s²+r²)-R1d)² = Rminor²
• (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2) -R4)^2 + (sqrt(s^2+r^2) -R1d)^2 = Rminor^2
------------------------------------
(((((R1a)R2)(R1b)R3)(R1c)R4)(R1d)Rm)
(((((24)12)(12)6)(8)3)(4)1) - ring torus diameter values
• (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -24)^2 +z^2) -12)^2 + (sqrt(w^2+v^2) -12)^2) -6)^2 + (sqrt(u^2+t^2) -12)^2) -3)^2 + (sqrt(s^2+r^2) -6)^2 = 1
------------------
XYrange = -48,+48
Zrange = -15,+15
------------------
3D Scans of the (((((I))(I))(I))(I)) 4x2x2x2 Array, Adjust ‘a’ to move ± in 4-space by [±Rn] value
----------------------------------------------------------------------------------------------------
• ((((())(I))(I))(I)) : [±R1A±R2] intercepts are 4 places of 4x2x2 array of 16 toruses
(sqrt((sqrt((sqrt((sqrt(0^2+a^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48
• (((((I))())(I))(I)) : [±R1B] intercepts are 2 places of 8x2x2 array of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(a^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
• (((((I))(I))())(I)) : [±R1C] intercepts are 2 places of 4x2x2x[R1 pair] of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(a^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
• (((((I))(I))(I))()) : [±R1D] intercepts are 2 places of 4x2x2x[Rm pair] of 32 toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt(a^2+0^2) -4)^2 = 1
-----------------------------------------------
3D Explore Functions , Single Translate+Rotate
-----------------------------------------------
• (((((a))(A))(I))(I)) : Single Trans+Rotate; {±R1A±R2 of 4x2x2} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(0^2+(x*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57
• (((((I))(a))(A))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -12)^2) -6)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -24 < a < 24
--- 0 < b  <1.57
• (((((I))(I))(A))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1C of 4x2x2x[R1-pair]}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt((z*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1
--- -13 < a < 13
--- 0 < b  <1.57
• (((((a))(I))(I))(A)) : {±R1A±R2 of 4x2x2} to {±R1D of 4x2x2x[Rm-pair]}
(sqrt((sqrt((sqrt((sqrt(0^2+(z*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57
• (((((A))(I))(a))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1A±R2 of 4x2x2}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1
--- -48 < a < 48
--- 0 < b  <1.57
--- Zrange = -15,15
• (((((I))(A))(I))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1B of 8x2x2}
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1
--- -24 < a < 24
--- 0 < b  <1.57
--- Zrange = -15,15
----------------------
Dual Translate+Rotate
----------------------
• (((((a))(Ac))(C))(I)) : Dual T+R , explores 4x2 dual-void rectangle array of ((((())(II))())(I))
(sqrt((sqrt((sqrt((sqrt((x*cos(b)-a*sin(b))^2) -24)^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+(y*cos(d)-c*sin(d))^2) -12)^2) -6)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1
--- [a,b,c,d] = [±24±12 ; 0 ; ±8 ; 1.57] : ((((())(II))())(I)) : [±R1A±R2]x[±R1C] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet]
• (((((A)c)(C))(a))(I)) : Dual T+R
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2 +(y*cos(d)-c*sin(d))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1
• (((((A))(Ca))(-))(Ic)) : Dual T+R, Preset [±R1C]=8, explores multiple hypervoids
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1
• (((((-)A)(Ca))(-))(Ic)) : Dual T+R, Preset Void Locations [±R1A]=24;[±R1C]=8, explores dual-hypervoid intercept region
(sqrt((sqrt((sqrt((sqrt(24^2) -24)^2 +(x*sin(b)+a*cos(b))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1
----------------------------------------------------------------
• Dual-Empty Hypervoid Intercepts <[For Navigational Purposes]>
(((((R1a)R2)(R1b))(R1c))(R1d))
(((((±24)±12)(±12))(±8))(±4))
------------------------------
(((((a)b)(c))(d))(e)) - [±a]x[±b]x[±c]x[±d]x[±e] : Hypervoid Locations of Ring Intercept Activity
-------------------------------------------------------------------------------------------------
(((((I)I)(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((I)I)))(I)), 2x1x2x[R1-quartet] of 16 tori
(((((a))(II))(b))(I)) - [±24±12]x[±8] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet] of 8 tori
(((((a))(I))(b))(II)) - [±24±12]x[±8] are 4x2 places of ((((I)))(II)), 1x1x8 column of 8 tori
(((((I))(a))(b))(II)) - [±12]x[±8] are 2x2 places of (((((I))))(II)), 1x1x16 column of 16 tori
(((((II))(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((II))))(I)), 1x1x2x[R1-octet] of 16 tori
(((((a)I)(b))(c))(II)) - [±24]x[±12]x[±8] are 2x2x2 places of ((((I)))(II)), 1x1x8 column of 8 tori
(((((a)I)(b))(I))(I)) - [±24]x[±12] are 2x2 places of ((((I))(I))(I)), 4x2x2 array of 16 tori
(((((a)I)(I))(b))(I)) - [±24]x[±8] are 2x2 places of ((((I)(I)))(I)), 2x2x2x[R1-pair] of 16 tori






10D (((((II)I)I)((II)I))I)(II)) - Tigritorus Diduotoric Torus , T4xC2xC3 , (((((R1a)R2a)R3)((R1b)R2b)R4)R5)(R1c)min)
----------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -R1A)^2 + z^2) -R2A)^2 + w^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1B)^2 + t^2) -R2B)^2) -R4)^2 + s^2) -R5)^2 + (sqrt(r^2 + q^2) -R1C)^2 -R6^2 = 0

• (((((I)))((I))))(I))
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -R1A)^2 + 0^2) -R2A)^2 + 0^2) -R3)^2 + (sqrt((sqrt(y^2 + 0^2) -R1B)^2 + 0^2) -R2B)^2) -R4)^2 + 0^2) -R5)^2 + (sqrt(z^2 + 0^2) -R1C)^2 -R6^2 = 0

diameter adj
(sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -a)^2 + 0^2) -(a/2))^2 + 0^2) -(a/4))^2 + (sqrt((sqrt(y^2 + 0^2) -(a/2))^2 + 0^2) -(a/4))^2) -c)^2 + 0^2) -d)^2 + (sqrt(z^2 + 0^2) -b)^2 = 1

-Set 80 cubes

-Set ranges:
0 < a < 30
0 < b < 3.5
0 < c < 4
0 < d < 1.75

XYbox = -60 / +60
Zrange = -10 / +10


in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: CalcPlot3D

Postby ICN5D » Sun Mar 15, 2015 10:12 pm

A new explore function library, for the non-toroidal hyperprisms, version 3/15/2015


Hyperprism Explore Function Library.txt
(35.21 KiB) Downloaded 34 times




Copy-Paste Text format, online copy:

Code: Select all

***********************************************************************************************
***********************************************************************************************
***********************************************************************************************
***********************************************************************************************
***********************************************************************************************
***********************************************************************************************
* Explore Function Library for Hyperprisms *
********************************************








II>IO : 5D Cylhemoctahedrinder Explore Functions
---------------------------------------------------
• I : 1D Line

|x| = a

• II : 2D Square ; extend Line along y

|x-y| + |x+y| = a

• II> : 3D Square Pyramid ; scale Square to a point along z

||x-y|+|x+y| + 3z| + |x-y|+|x+y| = a

• II>I : 4D Hemoctahedrinder ; extend Square Pyramid along w

|||x-y|+|x+y|+3z|+|x-y|+|x+y| - 4w| + |||x-y|+|x+y|+3z|+|x-y|+|x+y| + 4w| = a

• II>IO : 5D Cylhemoctahedrinder ; bisecting rotate Hemoctahedrinder around xyz, along w into v

|||x-y|+|x+y|+3z|+|x-y|+|x+y| - 4√(w²+v²)| + |||x-y|+|x+y|+3z|+|x-y|+|x+y| + 4√(w²+v²)| = a



• cube III , 3-plane XYW : ZV=0
abs(abs(abs(x-y)+abs(x+y) - 2a) + abs(x-y)+abs(x+y) - 3sqrt(z^2 + b^2)) + abs(abs(abs(x-y)+abs(x+y) - 2a) + abs(x-y)+abs(x+y) + 3sqrt(z^2 + b^2)) - 3.6^2
--- a : cube LxW has triangle symmetry
--- b : cube H has circular symm

• cylinder IIO , 3-plane XWV : YZ=0
abs(abs(abs(x-a)+abs(x+a) - b) + abs(x-a)+abs(x+a) - 3sqrt(y^2 + z^2)) + abs(abs(abs(x-a)+abs(x+a) - b) + abs(x-a)+abs(x+a) + 3sqrt(y^2 + z^2)) - 2.5^2
--- a : cylinder LxW has line symm
--- b : cylinder H has triangle symm

• square pyramid II> , 3-plane XYZ : WV=0
abs(abs(abs(x-y)+abs(x+y) - 2z) + abs(x-y)+abs(x+y) - 3sqrt(a^2 + b^2)) + abs(abs(abs(x-y)+abs(x+y) - 2z) + abs(x-y)+abs(x+y) + 3sqrt(a^2 + b^2)) - 3.5^2
--- a : pyramid has line symmetry
--- b : pyramid has line symmetry

• triangle prism I>I , 3-plane XZW : YV=0
abs(abs(abs(x-a)+abs(x+a) - 2y) + abs(x-a)+abs(x+a) - 3sqrt(z^2 + b^2)) + abs(abs(abs(x-a)+abs(x+a) - 2y) + abs(x-a)+abs(x+a) + 3sqrt(z^2 + b^2)) - 3.5^2
--- a : square atop triangle prism morph, with circular symm
--- b : triangle prism H has circular symm

• II>IO Rotate Function
----------------------------------------
abs(abs(abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 2(z*sin(b))) + abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 3sqrt((y*cos(a))^2 + (z*cos(b))^2)) + abs(abs(abs(x-(y*sin(a)))+abs(x+(y*sin(a))) - 2(z*sin(b))) + abs(x-(y*sin(a)))+abs(x+(y*sin(a))) + 3sqrt((y*cos(a))^2 + (z*cos(b))^2)) - 3.5^2
---[A,B] Rotate Positions
[0,0] - IIO cylinder
[1.57,0] - III cube
[1.57,1.57] - II> square pyramid
[0,1.57] - I>I triangle prism
GIF : http://hddb.teamikaria.com/dl/WTRHJPG4C6NZ8TR7MTR1RCJCYS.gif







IO>[I>] : 5D Contrianglinder Explore Functions
------------------------------------------------
• I : 1D Line ; extend point along x

|x| = a

• IO : 2D Circle ; bisecting rotate Circle around origin, along x into y

√(x²+y²) = a

• IO> : 3D Cone ; scale Circle to point along z

|√(x²+y²)+2z| + √(x²+y²) = a

• IO>I : 4D Coninder ; extend Cone along w

||√(x²+y²)+2z| + √(x²+y²) - w| + ||√(x²+y²)+2z| + √(x²+y²) + w| = a

• IO>[I>] : 5D Contrianglinder ; cartesian product of cone-xyz times triangle-wv

||√(x²+y²)+2z| + √(x²+y²) - ||w|+2v|-|w|| + ||√(x²+y²)+2z| + √(x²+y²) + ||w|+2v|+|w|| = a


3D Cross Sections of IO>[I>] along XYZWV
------------------------------------------
• ii>[I>] : Triangle Prism I>I , XY=0
abs(abs(sqrt(a^2+b^2) - 2x) + sqrt(a^2+b^2) - abs(abs(y)-2z) - abs(y)) + abs(abs(sqrt(a^2+b^2) - 2x) + sqrt(a^2+b^2) + abs(abs(y)-2z) + abs(y)) - 9.667
--- a,b : height has I> || I>I with circular symmetry

• Iii[I>] : Triangle Prism I>I , YZ=0
abs(abs(sqrt(x^2+a^2) - 2b) + sqrt(x^2+a^2) - abs(abs(y)-2z) - abs(y)) + abs(abs(sqrt(x^2+a^2) - 2b) + sqrt(x^2+a^2) + abs(abs(y)-2z) + abs(y)) - 9.667
--- a : z-height has circ symm
--- b : z-height has half height triangle symm

• Ii>[i>] : Triangle Prism I>I , YW=0
abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) - abs(abs(b)-2z) - abs(b)) + abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) + abs(abs(b)-2z) + abs(b)) - 9.667
--- a : y-width has smoothing collapse to line
--- b : z-height has full height triangle symm

• Ii>[Ii] : Triangle Prism I>I , YV=0
abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) - abs(abs(z)-2b) - abs(z)) + abs(abs(sqrt(x^2+a^2) - 2y) + sqrt(x^2+a^2) + abs(abs(z)-2b) + abs(z)) - 9.667
--- a : y-width has smoothing collapse to line
--- b : z-height has half height triangle symm

• IOi[i>] : Cylinder IOI , ZW=0
abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) - abs(abs(b)-2z) - abs(b)) + abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) + abs(abs(b)-2z) + abs(b)) - 9.667
--- a : I || IOI  , line atop cylinder symmetry
--- b : z-height has half-height triangular symmetry

• IOi[Ii] : Cylinder IOI , ZV=0
abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) - abs(abs(z)-2b) - abs(z)) + abs(abs(sqrt(x^2+y^2) - 2a) + sqrt(x^2+y^2) + abs(abs(z)-2b) + abs(z)) - 9.667
--- a : I || IOI , line atop cylinder with half-height triangle symm
--- b : IO || IOI , circle atop cylinder with half-height triangle symm

• IO>[ii] : Cone IO> , WV=0
abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) - abs(abs(a)-2b) - abs(a)) + abs(abs(sqrt(x^2+y^2) - 2z) + sqrt(x^2+y^2) + abs(abs(a)-2b) + abs(a)) - 9.667
--- a,b : cone has line symmetry

• IO>[ii] to Iii[I>] Rotate Function
------------------------------------
abs(abs(sqrt(x^2+(y*sin(a))^2) - 2(z*sin(b))) + sqrt(x^2+(y*sin(a))^2) - abs(abs((y*cos(a)))-2(z*cos(b))) - abs((y*cos(a)))) + abs(abs(sqrt(x^2+(y*sin(a))^2) - 2(z*sin(b))) + sqrt(x^2+(y*sin(a))^2) + abs(abs((y*cos(a)))-2(z*cos(b))) + abs((y*cos(a)))) - 10


[a,b] - 3D Midsection Positions
[0,0] - Triangle Prism I>I
[1.57,0] - Cylinder IOI
[1.57,1.57] - Cone IO>
[0,1.57] - Triangle Prism I>I
[0,0.785] - Square Pyramid II>
[0,2.355] - Tetrahedron I>>
[0,3.14] - Inverted I>I
GIF : http://hddb.teamikaria.com/dl/QWAWY48AF1XX0J35V6QAJ1Z27K.gif







• III , cube
-------------------
abs(abs(x - y) + abs(x + y) - 2z) + abs(abs(x - y) + abs(x + y) + 2z) - a

One-Axis Rotate of Cube in 3D
abs(abs((x*sin(a) + z*cos(a)) - y) + abs((x*sin(a) + z*cos(a)) + y) - 2(x*cos(a) - z*sin(a))) + abs(abs((x*sin(a) + z*cos(a)) - y) + abs((x*sin(a) + z*cos(a)) + y) + 2(x*cos(a) - z*sin(a))) - d

x -> (x*sin(a) + z*cos(a))
z -> (x*cos(a) - z*sin(a))

Two-Axis Rotate of Cube
abs(abs((x*sin(a) + z*cos(a)) - (y*sin(b) + (x*cos(a) - z*sin(a))*cos(b))) + abs((x*sin(a) + z*cos(a)) + (y*sin(b) + (x*cos(a) - z*sin(a))*cos(b))) - 2(y*cos(b) - (x*cos(a) - z*sin(a))*sin(b))) + abs(abs((x*sin(a) + z*cos(a)) - (y*sin(b) + (x*cos(a) - z*sin(a))*cos(b))) + abs((x*sin(a) + z*cos(a)) + (y*sin(b) + (x*cos(a) - z*sin(a))*cos(b))) + 2(y*cos(b) - (x*cos(a) - z*sin(a))*sin(b))) - 9.6

Two-Axis Rotate for any 3D shape: Replace x,y,z with,
x --> (x*sin(a) + z*cos(a))
y --> (y*sin(b) + (x*cos(a) - z*sin(a))*cos(b))
z --> (y*cos(b) - (x*cos(a) - z*sin(a))*sin(b))

abs(abs((x*sin(b) + a*cos(b)) - (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b)) + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) - 2(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs((x*sin(b) + a*cos(b)) - (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b)) + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + 2(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) = 9.6







• IIII , tesseract
-------------------------
||x-y|+|x+y| - |z-w|-|z+w|| + ||x-y|+|x+y| + |z-w|+|z+w|| = a

abs(abs(x-y)+abs(x+y) - abs(z-w)-abs(z+w)) + abs(abs(x-y)+abs(x+y) + abs(z-w)+abs(z+w)) = a

One axis rotate: X to A
x -> (x*sin(b) + a*cos(b))
a -> (x*cos(b) - a*sin(b))

Two-Axis Rotate: X to A , Y to A
x -> (x*sin(b) + a*cos(b))
y -> (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))
a -> (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))

Three-Axis Rotate: X to A , Y to A , Z to A
x -> (x*sin(b) + a*cos(b))
y -> (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))
z -> (z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))
a —> (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))

Triple-Axis Rotate of Tesseract
—————————————————
abs(abs((x*sin(b) + a*cos(b))-(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))+(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) - abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))-abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) + abs(abs((x*sin(b) + a*cos(b))-(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))+(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))+abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) = 10

-5 < a < 5
0 < b,c,d < 1.57

At a=0 midsection, perfect octahedron at angles [b,c,d] = [1.04667,0.96737,0.80879] . Use for sliding through corner first.

a=-1.1 : truncated tetrahedron, neat CRF shape








• IOOI> SPHERINDRONE
—————————————————————

• I : 1D Line

|x| = a

• IO : 2D Circle ; bisecting rotate Line around origin, along x into y

√(x²+y²) = a

• IOO : 3D Sphere ; bisecting rotate Circle around x, along y into z

√(x²+y²+z²) = a

• IOOI : 4D Spherinder ; extend Sphere along w

|√(x²+y²+z²) - w| + |√(x²+y²+z²) + w| = a

• IOOI> : 5D Spherindrone ; scale Spherinder to a point along v

||√(x²+y²+z²)-w|+|√(x²+y²+z²)+w| + 3v| + |√(x²+y²+z²)-w|+|√(x²+y²+z²)+w| = a

abs(abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) + 3v) + abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) = a

a = 8.5 , within XYZbox = -5,+5


3D Midsections of IOOI> along XYZWV

iiOI> : XY cut , II> sq pyramid
--------------------------------
abs(abs(sqrt(a^2+b^2+x^2) -y) + abs(sqrt(a^2+b^2+x^2) +y) - 3z) + abs(sqrt(a^2+b^2+x^2) -y) + abs(sqrt(a^2+b^2+x^2) +y) - 8.5
—— a : rounded collapse of pyramid height to line
—— b : rounded collapse of pyramid height to line

IOii> : ZW cut , IO> cone
--------------------------------
abs(abs(sqrt(x^2+y^2+a^2) -b) + abs(sqrt(x^2+y^2+a^2) +b) - 3z) + abs(sqrt(x^2+y^2+a^2) -b) + abs(sqrt(x^2+y^2+a^2) +b) - 8.5
—— a : rounded collapse of cone height
—— b : flat collapse of cone height to circle

IOiIi : ZV cut , IOI cylinder
--------------------------------
abs(abs(sqrt(x^2+y^2+a^2) -z) + abs(sqrt(x^2+y^2+a^2) +z) - 3b) + abs(sqrt(x^2+y^2+a^2) -z) + abs(sqrt(x^2+y^2+a^2) +z) - 8.5
—— a : diameter collapse to line
—— b : mid-height taper symmetry , -b shrink, +b expand

IOOii : WV cut , IOO sphere
--------------------------------
abs(abs(sqrt(x^2+y^2+z^2) -a) + abs(sqrt(x^2+y^2+z^2) +a) - 3b) + abs(sqrt(x^2+y^2+z^2) -a) + abs(sqrt(x^2+y^2+z^2) +a) - 8.5
—— a : linear symmetry, constant shape
—— b : mid-height taper symmetry , +b shrink, -b expand


IOOI> Rotation Function : YZ cut to WV cut, dual rotate
----------------------------------------------------------------
abs(abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 3(z*sin(b))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 8.5

[a,b] - 3D Midsection Positions
--------------------------------
[0,0] - IOO sphere
[1.57,0] - IOI cylinder
[1.57,1.57] - II> square pyramid
[0,1.57] - IO> cone








6D Sphone Diprism IOO>II
------------------------

IOO>II : 6D Sphone Diprism , cubic prism of the 4D spherical cone. Has infinite 4D sphones XYZW stacked within a square-segment VU

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IOO - 3D Sphere : bisecting rotate CIRCLE around stationary axis X, along Y into Z

√(x²+y²+z²) = a

IOO> - 4D Sphone : shrink SPHERE to point while extending along W

|√(x²+y²+z²) + 2w| + √(x²+y²+z²) = a

IOO>I - 5D Sphoninder : extend SPHONE along line into V

||√(x²+y²+z²)+2w| + √(x²+y²+z²) - 2v| + ||√(x²+y²+z²)+2w| + √(x²+y²+z²) + 2v| = a

IOO>II - 6D Sphone Diprism : extend SPHONINDER along line into U

||√(x²+y²+z²)+2w| + √(x²+y²+z²) - |v-u| - |v+u|| + ||√(x²+y²+z²)+2w| + √(x²+y²+z²) + |v-u| + |v+u|| = a

abs(abs(sqrt(x^2+y^2+z^2) + 2w) + sqrt(x^2+y^2+z^2) - abs(v-u) - abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) + 2w) + sqrt(x^2+y^2+z^2) + abs(v-u) + abs(v+u)) = a


3D midsections
—————————————
XYZWVU

IiiiII , iOiiII , iiOiII - cube III , YZW cut . Also identical to iii>II as the other cube-section along XYZ cut

abs(abs(sqrt(x^2+0^2+0^2) - 2*0) + sqrt(x^2+0^2+0^2) - abs(y-z) - abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) - 2*0) + sqrt(x^2+0^2+0^2) + abs(y-z) + abs(y+z)) = a


Iii>iI - triangle prism I>I , YZV cut

abs(abs(sqrt(x^2+0^2+0^2) - 2y) + sqrt(x^2+0^2+0^2) - abs(0-z) - abs(0+z)) + abs(abs(sqrt(x^2+0^2+0^2) - 2y) + sqrt(x^2+0^2+0^2) + abs(0-z) + abs(0+z)) = a


IOi>ii - cone IO> , ZVU cut

abs(abs(sqrt(x^2+y^2+0^2) - 2z) + sqrt(x^2+y^2+0^2) - abs(0-0) - abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) - 2z) + sqrt(x^2+y^2+0^2) + abs(0-0) + abs(0+0)) = a


IOiiiI - cylinder IOI , ZWV

abs(abs(sqrt(x^2+y^2+0^2) - 2*0) + sqrt(x^2+y^2+0^2) - abs(0-z) - abs(0+z)) + abs(abs(sqrt(x^2+y^2+0^2) - 2*0) + sqrt(x^2+y^2+0^2) + abs(0-z) + abs(0+z)) = a


IOOiii - sphere , WVU cut

abs(abs(sqrt(x^2+y^2+z^2) - 2*0) + sqrt(x^2+y^2+z^2) - abs(0-0) - abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) - 2*0) + sqrt(x^2+y^2+z^2) + abs(0-0) + abs(0+0)) = a


IOOiii to iii>II rotate function
---------------------------------

abs(abs(sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - 2(x*cos(a))) + sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - abs((y*cos(b))-(z*cos(c))) - abs((y*cos(b))+(z*cos(c)))) + abs(abs(sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) - 2(x*cos(a))) + sqrt((x*sin(a))^2+(y* sin(b))^2+(z*sin(c))^2) + abs((y*cos(b))-(z*cos(c))) + abs((y*cos(b))+(z*cos(c)))) = d

Set range to 0 < a,b,c < 1.57 for 90 deg turn in three independent circles of rotation, O < d < 15 for size scaling. Set XYZbox to -8, +8.

[a,b,c]-midsection positions

[1.57,1.57,1.57] - sphere IOO
[0,1.57,1.57] - cone IO>
[1.57,0,1.57] - cylinder IOI
[1.57,1.57,0] - cylinder IOI
[1.57,0,0] - cube III
[0,1.57,0] - triangle prism I>I
[0,0,1.57] - triangle prism I>I
[0,0,0] - cube III








• IOIO - Duocylinder
———————————
• I - 1D Line

|x| = a

• IO - 2D Circle, bisecting rotate Line around origin, along x into y

√(x²+y²) = a

• IOI - 3D Cylinder, extend Circle along z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

• IOIO - 4D Duocylinder, bisecting rotate Cylinder around xy, along z into w

|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

abs(sqrt(x^2 + y^2) - sqrt(z^2 + w^2)) + abs(sqrt(x^2 + y^2) + sqrt(z^2 + w^2)) - 3^2


3D Cross Section
abs(sqrt(x^2 + y^2) - sqrt(z^2 + a^2)) + abs(sqrt(x^2 + y^2) + sqrt(z^2 + a^2)) - 3^2

--- Makes cylinder with max height at A = 0 , min height at A = 4.5


Duocylinder Rotation
abs(sqrt(x^2 + (y*sin(a))^2) - sqrt(z^2 + (y*cos(a))^2)) + abs(sqrt(x^2 + (y*sin(a))^2) + sqrt(z^2 + (y*cos(a))^2)) - 3^2

Duocylinder Rotate + Translate
abs(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - sqrt(z^2 + (y*cos(b) - a*sin(b))^2)) + abs(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) + sqrt(z^2 + (y*cos(b) - a*sin(b))^2)) - 3^2








• I>IO - Cyltrianglinder
---------------------------------------
* - Point : Elementary starting shape

I - Line : extend Point along X

|x| = a

I> - Triangle : taper Line to point along Y

||x| + 2y| + |x| = a

I>I - Triangle Prism : extend Triangle along Z

|||x|+2y|+|x| - 2z| + |||x|+2y|+|x| - 2z| = a

I>IO = I>[IO] - Cyltrianglinder : bisecting rotate Triangle Prism around plane XY, along Z into W

|||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| = a

abs(abs(abs(x) + 2y) + abs(x) - 2sqrt(z^2 + w^2)) + abs(abs(abs(x) + 2y) + abs(x) + 2sqrt(z^2 + w^2)) - a


Frame Polynomial
I>IO • {3:IOI + I>(O)} • {3:IO + 3:I(O)} • 3:(O) •-• n

3D Cross Sections

• Cutting circular parameter to line , I>I with circular height
abs(abs(abs(x) - 2y) + abs(x) - sqrt(z^2 + a^2)) + abs(abs(abs(x) - 2y) + abs(x) + sqrt(z^2 + a^2)) - 6
--- makes I>I with circular height prism
--- min height at a = H , a = 2.9999 for 2D triangle
--- max height at a = 0

• Cutting triangular parameter to mid-length line , IOI with triangle symmetry
abs(abs(abs(x) - 2a) + abs(x) - sqrt(y^2 + z^2)) + abs(abs(abs(x) - 2a) + abs(x) + sqrt(y^2 + z^2)) - 6
--- makes IOI with triangular height prism
--- min height at a = -1.5 or -H/2
--- max height at a = 1.5 or +H/2


Rotation of I>IO
-----------------
abs(abs(abs(x) - 2(y*sin(a))) + abs(x) - sqrt(z^2 + (y*cos(a))^2)) + abs(abs(abs(x) - 2(y*sin(a))) + abs(x) + sqrt(z^2 + (y*cos(a))^2)) - 6

abs(abs(abs((x*sin(a))) - 2y) + abs((x*sin(a))) - sqrt(z^2 + (x*cos(a))^2)) + abs(abs(abs((x*sin(a))) - 2y) + abs((x*sin(a))) + sqrt(z^2 + (x*cos(a))^2)) - 6

Rotate  + Translate of I>IO , Y -> W
———————————————————————————
abs(abs(abs(x) + 2(y*sin(b) + a*cos(b))) + abs(x) - 2sqrt(z^2 + (y*cos(b) - a*sin(b))^2)) + abs(abs(abs(x) + 2(y*sin(b) + a*cos(b))) + abs(x) + 2sqrt(z^2 + (y*cos(b) - a*sin(b))^2)) - 9.7
— XYZbox = -5 / +5
--- a = -5.5 ~ 5.5
--- b = 0 ~ 1.57
--- good oblique angle is b = 0.445
--- put a = min/max height for IOI cut, then rotate b for cool stuff
--- For IOI cut, set a = -1.36 ~ +1.41

R+T X -> W
—————————
abs(abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) - 2sqrt(z^2 + (x*cos(b) - a*sin(b))^2)) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) + 2sqrt(z^2 + (x*cos(b) - a*sin(b))^2)) - 9.7

b -> b/57.3 , for angle in degrees , 0 < b < 90 , 45 steps, resolution = 2

Y -> W
abs(abs(abs(x) + 2(y*sin(b/57.3) + a*cos(b/57.3))) + abs(x) - 2sqrt(z^2 + (y*cos(b/57.3) - a*sin(b/57.3))^2)) + abs(abs(abs(x) + 2(y*sin(b/57.3) + a*cos(b/57.3))) + abs(x) + 2sqrt(z^2 + (y*cos(b/57.3) - a*sin(b/57.3))^2)) - 9.7

X -> W
abs(abs(abs((x*sin(b/57.3) + a*cos(b/57.3))) + 2y) + abs((x*sin(b/57.3) + a*cos(b/57.3))) - 2sqrt(z^2 + (x*cos(b/57.3) - a*sin(b/57.3))^2)) + abs(abs(abs((x*sin(b/57.3) + a*cos(b/57.3))) + 2y) + abs((x*sin(b/57.3) + a*cos(b/57.3))) + 2sqrt(z^2 + (x*cos(b/57.3) - a*sin(b/57.3))^2)) - 9.7









IO[IO]>I : 6D Duocylindroninder , prism of the pyramid of the 4D duocylinder. Has infinite 5D duocylindrical pyramids XYZWV stacked within line segment U

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IOI - 3D Cylinder : extend CIRCLE along line into Z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

IOIO = IO[IO] - 4D Duocylinder : bisecting rotate CYLINDER around stationary plane XY, along Z into W

|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

IO[IO]> - 5D Duocylindrone : shrink DUOCYLINDER to point while extending along V

||√(x²+y²)-√(z²+w²)| + |√(x²+y²)+√(z²+w²)| + 2v| + |√(x²+y²)-√(z²+w²)| + |√(x²+y²)+√(z²+w²)| = a

IO[IO]>I - 6D Duocylindroninder : extend DUOCYLINDRONE along line into U

|||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)|+2v| + |√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| - 4u| + |||√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)|+2v| + |√(x²+y²)-√(z²+w²)|+|√(x²+y²)+√(z²+w²)| + 4u| = a

———————

IO>I[IO] = IO>[IOI] : 6D Cubconinder , cartesian product of cone times cylinder. It has infinite cones XYZ stacked within a cylinder-segment WVU.


XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x²+y²) = a

IO> - 3D Cone : shrink CIRCLE to point while extending along Z

|√(x²+y²) + 2z| + √(x²+y²) = a

IO>I - 4D Coninder : extend CONE along line into W

||√(x²+y²)+2z| + √(x²+y²) - 2w| + ||√(x²+y²)+2z| + √(x²+y²) + 2w| = a

IO>IO = IO>[IO] - 5D Cylconinder : bisecting rotate CONINDER around stationary plane XYZ, along W into V

||√(x²+y²)+2z| + √(x²+y²) - 2√(w²+v²)| + ||√(x²+y²)+2z| + √(x²+y²) + 2√(w²+v²)| = a

IO>IOI = IO>[IOI] - Cubconinder : extend CYLCONINDER along line into U

||√(x²+y²)+2z| + √(x²+y²) - |√(w²+v²)-u| - |√(w²+v²)+u|| + ||√(x²+y²)+2z| + √(x²+y²) + |√(w²+v²)-u| + |√(w²+v²)+u|| = a









IO>I> : 5D Conindrone
—————————————————————

* - Point : Starting element

I - Line : extend Point along X

|x| = a

IO - Circle : bisecting rotate Line around point, along X into Y

√(x² + y²) = a

IO> - Cone : taper circle to point while extending along Z

|√(x²+y²) + 2z| + √(x²+y²) = a

IO>I - Coninder : extend Cone along W

||√(x²+y²)+2z|+√(x²+y²) - 2w| + ||√(x²+y²)+2z|+√(x²+y²) + 2w| = a

IO>I> - Conindrone : taper Coninder to point while extending along V

|||√(x²+y²)+2z|+√(x²+y²)-2w| + ||√(x²+y²)+2z|+√(x²+y²)+2w| + 4v| + ||√(x²+y²)+2z|+√(x²+y²)-2w| + ||√(x²+y²)+2z|+√(x²+y²)+2w| = a


abs(abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)-2w) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)+2w) + 4v) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)-2w) + abs(abs(sqrt(x^2+y^2)+2z)+sqrt(x^2+y^2)+2w) = a


3D Midsections of IO>I> along XYZWV
——————————————————————————

Circumradius = d = 10
XYZbox = -6 / +6


• ii>I> - square pyramid II> , XY cut

abs(abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)-2y) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)+2y) + 4z) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)-2y) + abs(abs(sqrt(a^2+b^2)+2x)+sqrt(a^2+b^2)+2y) = d

• IiiI> - square pyramid II> , YZ cut

abs(abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)+2y) + 4z) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2b)+sqrt(x^2+a^2)+2y) = d

• Ii>i> - tetrahedron I>> , YW cut

abs(abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2b) + 4y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2b) = d

• Ii>Ii - triangle prism I>I , YV cut

abs(abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2y) + 4b) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)-2y) + abs(abs(sqrt(x^2+a^2)+2z)+sqrt(x^2+a^2)+2y) = d

• IOii> - cone IO> , ZW cut

abs(abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2b) + 4z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2b) = d

• IOiIi - cylinder IOI , ZV cut

abs(abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2z) + 4b) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)-2z) + abs(abs(sqrt(x^2+y^2)+2a)+sqrt(x^2+y^2)+2z) = d

• IiiI> to IO>ii dual rotate function, y -> a , z -> b

abs(abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)-2(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)+2(y*sin(a))) + 4(z*sin(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)-2(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*cos(b)))+sqrt(x^2+(y*cos(a))^2)+2(y*sin(a))) = 10

• Ii>i> to IOiIi dual rotate, y -> a , z -> b

abs(abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)-2(z*cos(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)+2(z*cos(b))) + 4(y*sin(a))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)-2(z*cos(b))) + abs(abs(sqrt(x^2+(y*cos(a))^2)+2(z*sin(b)))+sqrt(x^2+(y*cos(a))^2)+2(z*cos(b))) = 10

— better rotate morphs

[a,b] - 3D Midsection Positions
[0,0] - cylinder IOI
[1.57,0] - square pyramid II>
[0,1.57] - cone IO>
[1.57,1.57] - tetrahedron I>>

• Ii>i> to IOiIi : y -> a , z -> b, Dual Rotate + Translate for full control over all angles and depth in 4D and 5D

abs(abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)-2(z*cos(d) - c*sin(d))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*cos(d) - c*sin(d))) + 4(y*sin(b) + a*cos(b))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)-2(z*cos(d) - c*sin(d))) + abs(abs(sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*sin(d) + c*cos(d)))+sqrt(x^2+(y*cos(b) - a*sin(b))^2)+2(z*cos(d) - c*sin(d))) = 10

-6 < a,c < +6
0 < b,d < 1.57

‘a’ slides, ‘b’ rotates in 4D
‘c’ slides, ‘d’ rotates in 5D

[b,d] - 3D Midsection Positions
[0,0] - cylinder IOI
[1.57,0] - square pyramid II>
[0,1.57] - cone IO>
[1.57,1.57] - tetrahedron I>>









Exploration of the 6D Duocylindrical Duotorus / Bi-Toroidal Prism [((II)I)((II)I)]
------------------------------------------------------------------------------------
[((II)I)((II)I)] - open toratope notation
IO(O)[IO(O)] - cartesian product of 2 orthogonal tori, (torus*torus)-prism, B^2 x S^1 x B^2 x S^1
IOIO[(O)(O)] - duocylinder bundle over the clifford torus, B^2 x B^2 x C^2

A few ways to build [((II)I)((II)I)] :


• I - 1D Line

|x| = a

• IO - 2D Circle, bisecting Line rotate around origin, along x into y

√(x²+y²) = a

• IO(O) - 3D Torus, shift Circle by -a along x, sweep along x into z

√((√(x²+z²)-a)² + y²) = b

√((√(x²+y²)-a)² + z²) = b

• IO(O)I - 4D Torinder, extend Torus along w

|√((√(x²+y²)-a)²+z²) - w| + |√((√(x²+y²)-a)²+z²) + w| = b

• IO(O)IO - 5D Cyltorinder, bisecting rotate Torinder around xyz, along w into v

|√((√(x²+y²)-a)²+z²) - √(w²+v²)| + |√((√(x²+y²)-a)²+z²) + √(w²+v²)| = b

• IO(O)[IO(O)] - 6D Bi-Toroidal Prism, shift Cyltorinder by -b along w, sweep along w into u

|√((√(x²+y²)-a)²+z²) - √((√(w²+u²)-b)²+v²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+u²)-b)²+v²)| = c

|√((√(x²+y²)-a)²+z²) - √((√(w²+v²)-b)²+u²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+v²)-b)²+u²)| = c




• I - 1D Line

|x| = a

• IO - 2D Circle, bisecting rotate Line around origin, along x into y

√(x²+y²) = a

• IOI - 3D Cylinder, extend Circle along z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

• IOIO - 4D Duocylinder, bisecting rotate Cylinder around xy, along z into w

|√(x²+y²) - √(z²+w²)| + |√(x²+y²) + √(z²+w²)| = a

• IOIO(O) - 5D Duocylindric Torus, shift Duocylinder by -a along x, sweep along x into v

|√((√(x²+v²)-a)²+y²) - √(z²+w²)| + |√((√(x²+v²)-a)²+y²) + √(z²+w²)| = b

|√((√(x²+y²)-a)²+z²) - √(w²+v²)| + |√((√(x²+y²)-a)²+z²) + √(w²+v²)| = b

• IOIO[(O)(O)] - 6D Duocylindric Tiger, shift Duocylindric Torus by -b along w, sweep along w into u

|√((√(x²+y²)-a)²+z²) - √((√(w²+u²)-b)²+v²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+u²)-b)²+v²)| = c

|√((√(x²+y²)-a)²+z²) - √((√(w²+v²)-b)²+u²)| + |√((√(x²+y²)-a)²+z²) + √((√(w²+v²)-b)²+u²)| = c

--------------------------------------
|√((√(x²+y²)-R1a)²+z²) - √((√(w²+v²)-R1b)²+u²)| + |√((√(x²+y²)-R1a)²+z²) + √((√(w²+v²)-R1b)²+u²)| = Rminor

abs(sqrt((sqrt(x^2+y^2)-R1a)^2+z^2) - sqrt((sqrt(w^2+v^2)-R1b)^2-u^2)) + abs(sqrt((sqrt(x^2+y^2)-R1a)^2+z^2) + sqrt((sqrt(w^2+v^2)-R1b)^2+u^2)) = Rm

abs(sqrt((sqrt(x^2+y^2)-3)^2+z^2) - sqrt((sqrt(w^2+v^2)-3)^2-u^2)) + abs(sqrt((sqrt(x^2+y^2)-3)^2+z^2) + sqrt((sqrt(w^2+v^2)-3)^2+u^2)) = 1


3D Slices

XYZbox = -7 / +7

• [((II))((I))] : Vertical stack of 2 square tori
abs((sqrt(x^2+y^2) -3)^2 + 0^2 - (sqrt(z^2+0^2) -3)^2 - 0^2) + abs((sqrt(x^2+y^2) -3)^2 + 0^2 + (sqrt(z^2+0^2) -3)^2 + 0^2) = 3

• [((I)I)((I))] : 4 cylinders in 2x2 square array
abs((sqrt(x^2+0^2) -3)^2 + y^2 - (sqrt(z^2+0^2) -3)^2 - 0^2) + abs((sqrt(x^2+0^2) -3)^2 + y^2 + (sqrt(z^2+0^2) -3)^2 + 0^2) = 3

-------------------------------------------

3D Explore Functions

• [((IY))((Iy))] : Single Rotate between stacks of square tori
abs((sqrt(x^2+(y*sin(a))^2) -3)^2 + 0^2 - (sqrt(z^2+(y*cos(a))^2) -3)^2 - 0^2) + abs((sqrt(x^2+(y*sin(a))^2) -3)^2 + 0^2 + (sqrt(z^2+(y*cos(a))^2) -3)^2 + 0^2) = 3

• [((XY)z)((Zx)y)] : Triple Rotate between stacks of sq tori, four cylinders in square, and hypervoids
abs((sqrt((x*sin(a))^2+(y*sin(b))^2) -3)^2 + (z*cos(c))^2 - (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 - (y*cos(b))^2) + abs((sqrt((x*sin(a))^2+(y*sin(b))^2) -3)^2 + (z*cos(c))^2 + (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 + (y*cos(b))^2) = 3
—— Cage of Bi-toroidal prism at a=0.785 ; b,c=1.57 // Direct analogue to Cage of Duotorus Tiger (((IO))((IO)))
—— 0 < a,b,c < 1.57

• [((Xz)Y)((Zx)y)] : Triple Rotate between duocylinder sections in square array and hypervoids
abs((sqrt((x*sin(a))^2+(z*cos(c))^2) -3)^2 + (y*sin(b))^2 - (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 - (y*cos(b))^2) + abs((sqrt((x*sin(a))^2+(z*cos(c))^2) -3)^2 + (y*sin(b))^2 + (sqrt((z*sin(c))^2+(x*cos(a))^2) -3)^2 + (y*cos(b))^2) = 4
—— Very interesting structure at a,c=1.157 ; b=0.785
—— 0 < a,b,c < 1.57

• [((AC)c)((Ia))] : Dual Translate+Rotate
abs((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2) -3)^2 + (y*cos(d)-c*sin(d))^2 - (sqrt(z^2+(x*cos(b)-a*sin(b))^2) -3)^2 - 0^2) + abs((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2) -3)^2 + (y*cos(d)-c*sin(d))^2 + (sqrt(z^2+(x*cos(b)-a*sin(b))^2) -3)^2 + 0^2) = 3
—— -7 < a,c < +7
—— 0 < b,d < 1.57







5D Cubtrianglinder I>IIO
-----------------------------------------
• I : 1D Line

|x| = a

• I> : 2D Triangle ; scale Line to a point across y

||x| +2y| + |x| = a

• I>I : 3D Triangle Prism ; extend Triangle along z

|||x|+2y|+|x| - 2z| + |||x|+2y|+|x| + 2z| = a

• I>II = II[I>] : 4D Triangle Diprism ; extend Triangle Prism along w

|||x|+2y|+|x| - |z-w|-|z+w|| + |||x|+2y|+|x| + |z-w|+|z+w|| = a

• I>IIO = IOI[I>] : 5D Cubtrianglinder ; bisecting rotate Triangle Diprism around xyw, along z into v

|||x|+2y|+|x| - |√(z²+v²)-w|-|√(z²+v²)+w|| + |||x|+2y|+|x| + |√(z²+v²)-w|+|√(z²+v²)+w|| = a

|||x|+2y|+|x| - |√(z²+w²)-v|-|√(z²+w²)+v|| + |||x|+2y|+|x| + |√(z²+w²)-v|+|√(z²+w²)+v|| = a

||√(x²+y²)-z|-|√(x²+y²)+z| - ||w|+2v|-|w|| + ||√(x²+y²)-z|-|√(x²+y²)+z| + ||w|+2v|+|w|| = a

abs(abs(sqrt(x^2+y^2)-z)-abs(sqrt(x^2+y^2)+z) - abs(abs(w)+2v)-abs(w)) + abs(abs(sqrt(x^2+y^2)-z)-abs(sqrt(x^2+y^2)+z) + abs(abs(w)+2v)+abs(w)) = a


• I : 1D Line

|x| = a

• I> : 2D Triangle ; scale Line to a point across y

||x| +2y| + |x| = a

• I>I : 3D Triangle Prism ; extend Triangle along z

|||x|+2y|+|x| - 2z| + |||x|+2y|+|x| + 2z| = a

• I>IO : 4D Cyltrianglinder ; bisecting rotate Triangle Prism around xy, along z into w

|||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| = a

• I>IOI : 5D Cubtrianglinder ; extend Cyltrianglinder along v

||||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| - 4v| + ||||x|+2y|+|x| - 2√(z²+w²)| + |||x|+2y|+|x| + 2√(z²+w²)| + 4v| = a

abs(abs(abs(abs(x)+2y)+abs(x) - 2sqrt(z^2+w^2)) + abs(abs(abs(x)+2y)+abs(x) + 2sqrt(z^2+w^2)) - 4v) + abs(abs(abs(abs(x)+2y)+abs(x) - 2sqrt(z^2+w^2)) + abs(abs(abs(x)+2y)+abs(x) + 2sqrt(z^2+w^2)) + 4v) = a









Exploring the 5D Cubindrone IOII> : 5D pyramid of the 4D prism of the 3D Cylinder
---------------------------------------------------------------------------------
• I : 1D Line

|x| = a

• IO : 2D Circle ; bisecting rotate Line around origin, along x into y

√(x²+y²) = a

• IOI : 3D Cylinder ; extend Circle along z

|√(x²+y²) - z| + |√(x²+y²) + z| = a

• IOII = IO[II] : 4D Cubinder ; extend Cylinder along w

|2√(x²+y²) - |z-w|-|z+w|| + |2√(x²+y²) + |z-w|+|z+w|| = a

• IOII> = IO[II]> : 5D Cubindrone ; scale Cubinder to a point along v

||2√(x²+y²)-|z-w|-|z+w|| + |2√(x²+y²)+|z-w|+|z+w|| + 6v| + |2√(x²+y²)-|z-w|-|z+w|| + |2√(x²+y²)+|z-w|+|z+w|| = a

abs(abs(2sqrt(x^2+y^2)-abs(z-w)-abs(z+w)) + abs(2sqrt(x^2+y^2)+abs(z-w)+abs(z+w)) + 6v) + abs(2sqrt(x^2+y^2)-abs(z-w)-abs(z+w)) + abs(2sqrt(x^2+y^2)+abs(z-w)+abs(z+w)) = a


3D Midsections of IOII> along XYZWV
-------------------------------------

XYZbox = -7,+7

• XYZWV

• iiII> : XY=0 ; II> square pyramid
abs(abs(2sqrt(0^2+0^2)-abs(x-y)-abs(x+y)) + abs(2sqrt(0^2+0^2)+abs(x-y)+abs(x+y)) + 6z) + abs(2sqrt(0^2+0^2)-abs(x-y)-abs(x+y)) + abs(2sqrt(0^2+0^2)+abs(x-y)+abs(x+y)) = 17

• IiIIi : YV=0 ; III cube
abs(abs(2sqrt(x^2+0^2)-abs(y-z)-abs(y+z)) + abs(2sqrt(x^2+0^2)+abs(y-z)+abs(y+z)) + 6*0) + abs(2sqrt(x^2+0^2)-abs(y-z)-abs(y+z)) + abs(2sqrt(x^2+0^2)+abs(y-z)+abs(y+z)) = 17

• IOii> : ZW=0 ; IO> cone
abs(abs(2sqrt(x^2+y^2)-abs(0-0)-abs(0+0)) + abs(2sqrt(x^2+y^2)+abs(0-0)+abs(0+0)) + 6z) + abs(2sqrt(x^2+y^2)-abs(0-0)-abs(0+0)) + abs(2sqrt(x^2+y^2)+abs(0-0)+abs(0+0)) = 17

• IOIii : WV=0 ; IOI cylinder
abs(abs(2sqrt(x^2+y^2)-abs(z-0)-abs(z+0)) + abs(2sqrt(x^2+y^2)+abs(z-0)+abs(z+0)) + 6*0) + abs(2sqrt(x^2+y^2)-abs(z-0)-abs(z+0)) + abs(2sqrt(x^2+y^2)+abs(z-0)+abs(z+0)) = 17


Exploratory Function of the 3-Plane, in 4-Space and 5-Space

• IiIi> --> IOiIi : Y -> Z , W -> V : Dual Rotate+Translate
abs(abs(2sqrt(x^2+y^2)-abs(z-w)-abs(z+w)) + abs(2sqrt(x^2+y^2)+abs(z-w)+abs(z+w)) + 6v) + abs(2sqrt(x^2+y^2)-abs(z-w)-abs(z+w)) + abs(2sqrt(x^2+y^2)+abs(z-w)+abs(z+w)) = a

y = (y*sin(b)+a*cos(b))
z = (y*cos(b)-a*sin(b))

w = (z*sin(d)+c*cos(d))
v = (z*cos(d)-c*sin(d))

abs(abs(2sqrt(x^2+(y*sin(b)+a*cos(b))^2)-abs((y*cos(b)-a*sin(b))-(z*sin(d)+c*cos(d)))-abs((y*cos(b)-a*sin(b))+(z*sin(d)+c*cos(d)))) + abs(2sqrt(x^2+(y*sin(b)+a*cos(b))^2)+abs((y*cos(b)-a*sin(b))-(z*sin(d)+c*cos(d)))+abs((y*cos(b)-a*sin(b))+(z*sin(d)+c*cos(d)))) + 6(z*cos(d)-c*sin(d))) + abs(2sqrt(x^2+(y*sin(b)+a*cos(b))^2)-abs((y*cos(b)-a*sin(b))-(z*sin(d)+c*cos(d)))-abs((y*cos(b)-a*sin(b))+(z*sin(d)+c*cos(d)))) + abs(2sqrt(x^2+(y*sin(b)+a*cos(b))^2)+abs((y*cos(b)-a*sin(b))-(z*sin(d)+c*cos(d)))+abs((y*cos(b)-a*sin(b))+(z*sin(d)+c*cos(d)))) = 17

Slide in 4D,5D
-6 < a,c < 6

Rotate in 4D,5D
0 < b,d < 1.57

[b,d] - 3D Midsection Positions
[0,0] - Square Pyramid II>
[1.57,0] - Cone IO>
[1.57,1.57] - Cylinder IOI
[0,1.57] - Cube III









4D Spherinder IOOI
-------------------------------------------
I - 1D Line : extend Point along X

|x| = a

IO - 2D Circle : bisecting rotate Line around origin, along X into Y

√(x²+y²) = a

IOO - 3D Sphere : bisecting rotate Circle around X, along Y into Z

√(x²+y²+z²) = a

IOOI - 4D Spherinder : extend Sphere along W

|√(x²+y²+z²) - w| + |√(x²+y²+z²) + w| = a

abs(sqrt(x^2+y^2+z^2)-w) + abs(sqrt(x^2+y^2+z^2)+w) = a


3D midsections of IOOI along XYZW

i = dimension set to zero

IOiI - cylinder IOI , Z=0
abs(sqrt(x^2+y^2+0^2)-z) + abs(sqrt(x^2+y^2+0^2)+z) = a

IOOi - sphere IOO , W=0
abs(sqrt(x^2+y^2+z^2)-0) + abs(sqrt(x^2+y^2+z^2)+0) = a


3D Explore Functions

Rotate Function z -> w
abs(sqrt(x^2+y^2+(z*sin(a))^2)-(z*cos(a))) + abs(sqrt(x^2+y^2+(z*sin(a))^2)+(z*cos(a))) = 7

Translate + Rotate Function z -> w
abs(sqrt(x^2+y^2+(z*sin(b)+a*cos(b))^2)-(z*cos(b)-a*sin(b))) + abs(sqrt(x^2+y^2+(z*sin(b)+a*cos(b))^2)+(z*cos(b)-a*sin(b))) = 7

in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: CalcPlot3D

Postby ICN5D » Tue Sep 15, 2015 11:53 pm

Here's a good collection of explore functions that move a shape around in a higher dimension. These are the best I've used, especially the X,Y,Z -> [xyz] function, which allows for a much larger and wider variety of topology changes.

Translate 'a' with single rotation 'b' of sliding direction

x = (x*sin(b)+a*cos(b))
a = (x*cos(b)-a*sin(b))

---

Translate 'a' with 2 rotation parameters 'b,c' of slide direction

x = (x*sin(b)+a*cos(b))
y = (y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))
[xy] = (y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))

---

Translate 'a' with 3 rotation parameters 'b,c,d' of slide direction, and a rotation switch of [xyz] endpoint using 't'

x = (x*sin(b)+a*cos(b))
y = (y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))
z = (z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))
[xyzT] = ((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))*sin(t))
[xyzt] = ((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))*cos(t))

---

Translate 'a' with 4 rotate parameters 'b,c,d,t' of slide direction. To utilize this function to its fullest potential, CalcPlot will need 6 adjustable parameters. It has 5 for now, but we can hard-set W to a specific value, if exploring a 5D array of intercepts.

x = (x*sin(b)+v*cos(b))
y = (y*sin(c)+(x*cos(b)-v*sin(b))*cos(c))
z = (z*sin(d)+(y*cos(c)-(x*cos(b)-v*sin(b))*sin(c))*cos(d))
w = (w*sin(t)+(z*cos(d)-(y*cos(c)-(x*cos(b)-v*sin(b))*sin(c))*sin(d))*cos(t))
v = (w*cos(t)-(z*cos(d)-(y*cos(c)-(x*cos(b)-v*sin(b))*sin(c))*sin(d))*sin(t))
in search of combinatorial objects of finite extent
ICN5D
Pentonian
 
Posts: 1044
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

Re: CalcPlot3D

Postby ICN5D » Wed Sep 16, 2015 12:05 am

I also went all out lately, with 8,9, and 10D toratope exploration. I broke into the 10D tori that make 5D arrays of intercepts, quite possibly the most complex under 12D. I ended up here after setting out to see what 5D and 6D toratopes spaced apart in 4D arrays would look like, just for the heck of it. A few of the 8D ones have been posted before, but not the lot of them.

Code: Select all

############################################################################################

The Five Fundamental 8D Toratopes that intercept as 4D Arrays of 4D Toratopes

((II)(II)(II)(II)) - Tetratiger : 2x2x2x2 array of 16 glomes (IIII)
(sqrt((x*sin(b)+a*cos(b))^2)-3)^2 + (sqrt((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2)-3)^2 + (sqrt((z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2)-3)^2 + (sqrt((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)-3)^2 = 1

(((II)(II)(II))(II)) - Torispheric Quattrotorus : 2x2x2x2 array of 16 torispheres ((III)I)
(sqrt((sqrt((x*sin(b)+a*cos(b))^2)-6)^2+(sqrt((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2)-6)^2+(sqrt((z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2)-6)^2)-3)^2 + (sqrt((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)-6)^2 = 1

(((II)(II))(II)(II)) - Spheritoric Quattrotorus : 2x2x2x2 array of 16 spheritoruses ((II)II)
(sqrt((sqrt((x*sin(b)+a*cos(b))^2)-6)^2+(sqrt((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2)-6)^2)-3)^2+(sqrt((z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2)-6)^2+(sqrt((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)-6)^2 = 1

((((II)(II))(II))(II)) - Triple Tiger : 2x2x2x2 array of 16 ditoruses (((II)I)I)
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-8)^2+(sqrt((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2)-8)^2)-4)^2+(sqrt((z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2)-8)^2)-2)^2+(sqrt((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)-8)^2 = 1

(((II)(II))((II)(II)))  - Duotiger-tiger : 2x2x2x2 array of 16 tigers ((II)(II))
(sqrt((sqrt((x*sin(b)+a*cos(b))^2)-6)^2+(sqrt((y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2)-6)^2)-2)^2+(sqrt((sqrt((z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2)-6)^2+(sqrt((z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)-6)^2)-2)^2 = 1

Ranges:
-19 < a < 19
pi/3 < b < pi/2
0.9554 < c < pi/2
pi/4 < d < pi/2

Passing the 4D Array Through...
Corner First: b=pi/3 , c=0.9554 , d=pi/4 , animate -19 < a < 19
Line First: b=pi/2 , c=0.9554 , d=pi/4  , animate -19 < a < 19
Square First: b=pi/2 , c=pi/2 , d=pi/4  , animate -19 < a < 19
Cube First: b=pi/2 , c=pi/2 , d=pi/2  , animate -19 < a < 19

############################################################################################







8D ((((II)I)I)(((II)I)I)) : Duoditorus Tiger , S1xC2xC2xC2
-------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 +w^2) -R3a)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -R1b)^2 +t^2) -R2b)^2 +s^2) -R3b)^2 = Rminor^2

(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +z^2) -5)^2 +w^2) -2.5)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -10)^2 +t^2) -5)^2 +s^2) -2.5)^2 = 0.5
• ((((II)))(((I))))
(sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+0^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
— XYZbox = -26 / +26
-- 42 cubes
****
• ((((IA)))(((Ia)))) : y -> A,a
(sqrt((sqrt((sqrt(x^2+(y*sin(b) + a*cos(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5
• ((((AY)c)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*((sin(c))*(sin(d))))^2) -10)^2 +(y*cos(c))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.75
• ((((Ac)Y)d)(((Ia))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(c))^2) -10)^2 +(y*((sin(c))*(sin(d))))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.75
****
• ((((IY)a)b)(((Ic)d)t))
(sqrt((sqrt((sqrt(x^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -10)^2 +(y*cos(a))^2) -5)^2 +(y*cos(b))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(c))^2) -10)^2 +(y*cos(d))^2) -5)^2 +(y*cos(t))^2) -2.5)^2 = 0.75
• ((((Ia)Y)b)(((Ic)d)t))
(sqrt((sqrt((sqrt(x^2+(y*cos(a))^2) -10)^2 +(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -5)^2 +(y*cos(b))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(c))^2) -10)^2 +(y*cos(d))^2) -5)^2 +(y*cos(t))^2) -2.5)^2 = 0.75
• ((((Ia)b)Y)(((Ic)d)t))
(sqrt((sqrt((sqrt(x^2+(y*cos(a))^2) -10)^2 +(y*cos(b))^2) -5)^2 +(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(c))^2) -10)^2 +(y*cos(d))^2) -5)^2 +(y*cos(t))^2) -2.5)^2 = 0.75
• ((((Ia)b)c)(((IY)d)t))
(sqrt((sqrt((sqrt(x^2+(y*cos(a))^2) -10)^2 +(y*cos(b))^2) -5)^2 +(y*cos(c))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -10)^2 +(y*cos(d))^2) -5)^2 +(y*cos(t))^2) -2.5)^2 = 0.75
• ((((Ia)b)c)(((Id)Y)t))
(sqrt((sqrt((sqrt(x^2+(y*cos(a))^2) -10)^2 +(y*cos(b))^2) -5)^2 +(y*cos(c))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(d))^2) -10)^2 +(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -5)^2 +(y*cos(t))^2) -2.5)^2 = 0.75
• ((((Ia)b)c)(((Id)t)Y))
(sqrt((sqrt((sqrt(x^2+(y*cos(a))^2) -10)^2 +(y*cos(b))^2) -5)^2 +(y*cos(c))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(d))^2) -10)^2 +(y*cos(t))^2) -5)^2 +(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -2.5)^2 = 0.75
****
• ((((Da)I)b)(((Zd)c))) - X:D(t)->d(t) , Z:Z->a,b,c ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(z*cos(a))^2)-10)^2+y^2)-5)^2+(z*cos(b))^2)-2.5)^2+(sqrt((sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))))^2+(x*cos(t)-d*sin(t))^2)-10)^2+(z*cos(c))^2)-5)^2)-2.5)^2 = 0.75
• ((((Da)I)b)(((Zd))c)) - X:D(t)->d(t) , Z:Z->a,b,c ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(z*cos(a))^2)-10)^2+y^2)-5)^2+(z*cos(b))^2)-2.5)^2+(sqrt((sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))))^2+(x*cos(t)-d*sin(t))^2)-10)^2)-5)^2+(z*cos(c))^2)-2.5)^2 = 0.75
---
• ((((Da)Y)b)(((Id)c))) - X:D(t)->d(t) , Y:Y->a,b,c ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*cos(a))^2)-10)^2+(y*((sin(a))*(sin(b))*(sin(c))))^2)-5)^2+(y*cos(b))^2)-2.5)^2+(sqrt((sqrt((sqrt(z^2+(x*cos(t)-d*sin(t))^2)-10)^2+(y*cos(c))^2)-5)^2)-2.5)^2 = 0.75
• ((((D)Y)a)(((Id)b)c)) - X:D(t)->d(t) , Y:Y->a,b,c ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2)-10)^2+(y*((sin(a))*(sin(b))*(sin(c))))^2)-5)^2+(y*cos(a))^2)-2.5)^2+(sqrt((sqrt((sqrt(z^2+(x*cos(t)-d*sin(t))^2)-10)^2+(y*cos(b))^2)-5)^2+(y*cos(c))^2)-2.5)^2 = 0.75
---
• ((((DY)a)d)(((I)b)c)) - X:D(t)->d(t) , Y:Y->a,b,c ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*((sin(a))*(sin(b))*(sin(c))))^2)-10)^2+(y*cos(a))^2)-5)^2+(x*cos(t)-d*sin(t))^2)-2.5)^2+(sqrt((sqrt((sqrt(z^2+0^2)-10)^2+(y*cos(b))^2)-5)^2+(y*cos(c))^2)-2.5)^2 = 0.75
---
• ((((DY))z)(((Zd)a)b)) - X:D(t)->d(t) , Y:Y->a,b , Z:Z->z(c) ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*((sin(a))*(sin(b))))^2) -10)^2 +0^2) -5)^2 +(z*cos(c))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(c))^2+(x*cos(t)-d*sin(t))^2) -10)^2 +(y*cos(a))^2) -5)^2 +(y*cos(b))^2) -2.5)^2 = 0.75
• ((((DY)a)b)(((Zd))y)) - X:D(t)->d(t) , Y:Y->y(c) , Z:Z->a,b ; 0 < a,b,c,t < 1.57 , -28 < d < 28
(sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*sin(c))^2) -10)^2 +(z*cos(a))^2) -5)^2 +(z*cos(b))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*((sin(a))*(sin(b))))^2+(x*cos(t)-d*sin(t))^2) -10)^2 +0^2) -5)^2 +(y*cos(c))^2) -2.5)^2 = 0.75
-- set a=1.57;b,c=0;t=0.785 : animate d for the diagonal scan of the 4x4 square array of 16 tigers
---
• ((((A)C)z)(((Zc)a))) - X:A(b)->a(b) , Y:C(d)->c(d) , Z:Z->z(t) ; 0 < b,d,t < 1.57 , -28 < a,c < 28
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2)-10)^2+(y*sin(d)+c*cos(d))^2)-5)^2+(z*cos(t))^2)-2.5)^2+(sqrt((sqrt((sqrt((z*sin(t))^2+(y*cos(d)-c*sin(d))^2)-10)^2+(x*cos(b)-a*sin(b))^2)-5)^2+0^2)-2.5)^2 = 0.75







8D ((((II)(II))I)((II)I)) : [Toratiger x Torus]-tiger , S1xC2xS1xS1xC2 , T2xC2xC3 , S1xC2xS1xC3
------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R3)^2 +v^2) -R4)^2 + (sqrt((sqrt(u^2+t^2) -R1c)^2 +s^2) -R2)^2 = Rm^2
• Trace Array Diameter Adjustment Equation
(sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) -b)^2 +0^2) -(b/2))^2 = 1
• ((((I)(I)))((I))) : 2x2x4x[R1 pair] array of 32 toruses
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
—— XYZbox = -22 / +22
• ((((I)(I)))((I))) : standard explore function slate
(sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1
• ((((A)(Ca)))((Ic))) : X -> A,a / Y -> C,c
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-9)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-9)^2)-4.5)^2)-2)^2+(sqrt((sqrt(z^2+(y*cos(d)-c*sin(d))^2)-6.5)^2)-3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((Ic)(A))a)((C))) : Y -> A,a / Z -> C,c
(sqrt((sqrt((sqrt(x^2+(z*cos(d)-c*sin(d))^2)-9)^2+(sqrt((y*sin(b)+a*cos(b))^2)-9)^2)-4.5)^2+(y*cos(b)-a*sin(b))^2)-2)^2+(sqrt((sqrt((z*sin(d)+c*cos(d))^2)-6.5)^2)-3.25)^2 = 1
—— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20
• ((((A)(I))c)((C)a)) : X -> A,a / Z -> C,c
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-9)^2+(sqrt(y^2)-9)^2)-4.5)^2+(z*cos(d)-c*sin(d))^2)-2)^2+(sqrt((sqrt((z*sin(d)+c*cos(d))^2)-6.5)^2(x*cos(b)-a*sin(b))^2)-3.25)^2 = 1







8D ((((II)I)(II))((II)I)) - Double Tiger A,C-Duotorus , T2xC3xC2 = S1x[(S1xC2xS1)*T2] , ((((R1a)R2a)(R1b)R3)((R1c)R2b)r)
-----------------------------------------------------------------------------------------------------------------------------------------------------------
(II) - (1)
((II)I) - ((2.5)1)
((II)(II)) - ((2.5)(2.5)1)
(((II)I)(II)) - (((5)2.5)(2.5)1)
(((II)I)((II)I)) - (((5)2.5)((5)2.5)1)
(((II)(II))((II)I)) - (((5)(5)2.5)((5)2.5)1)
((((II)I)(II))((II)I)) - ((((10)5)(5)2.5)((5)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + z^2) -5)^2 + (sqrt(w^2 + v^2) -5)^2) -2.5)^2 + (sqrt((sqrt(u^2 + t^2) -5)^2 + s^2) -2.5)^2 = 1

• ((((I))(I))((I))) : 32 torus intercepts in 4x2x4 brick array
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt(y^2 + 0^2) -5)^2) -2.5)^2 + (sqrt((sqrt(z^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)[xyzT])(Y[xyzt]))((Z)))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2) -2.5)^2 = 1

• ((((X)[xyzT])(Y))((Z[xyzt])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 = 1

• ((((X)[xyzT])(Y))((Z)[xyzt]))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1
---
• ((((X[xyzT]))(Y))((Z)[xyzt]))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

^^^ Has very awesome morphs at a=5 , c=pi/4 , d=0 , t=0 , animate 0 < b < 2pi . Shows a momentary (((I)I)((I))) 2x1x4 square in two flip-flopped positions, with a very complex topology change with high level of symmetry.
---
• ((((X[xyzT]))(Y))((Z[xyzt])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 = 1

^^^ Has awesome morphs analogous to above ((((X[xyzT]))(Y))((Z)[xyzt])) at a=5 , c=pi/4 , d=pi/2 , t=0 , animate 0 < b < 2pi . Shows a momentary (((II))((I))) 1x1x4x[R1 pair] column in two flip-flopped positions, between complex topo changes.
---







8D (((((II)I)I)I)((II)I)) - Duotorus-Tiger Ditorus , S1 x C2 x C2 x T2
-----------------------------------------------------------------------------
• (((((II)I)I)I)((II)I))
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 +w^2) -R3)^2 +v^2) -R4)^2 + (sqrt((sqrt(u^2+t^2) -R1b)^2 +s^2) -R2b)²^2 = Rmin^2

• (((((20)10)5)2.5)((5)2.5)1)
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -20)^2 +z^2) -10)^2 +w^2) -5)^2 +v^2) -2.5)^2 + (sqrt((sqrt(u^2+t^2) -5)^2 +s^2) -2.5)^2 = 1.25

• (((((II))))((I)))
(sqrt((sqrt((sqrt((sqrt(x^2+y^2)-20)^2+0^2)-10)^2+0^2)-5)^2+0^2)-2.5)^2+(sqrt((sqrt(z^2+0^2)-5)^2+0^2)-2.5)^2 = 1.25

3D Intersections, adjust a,b,c,d,t to move in 5D cut array
-----------------------------------------------------------
XYZbox = -42 , +42
51 Cubes

• (((((I))))((I)I))
(sqrt((sqrt((sqrt((sqrt(x^2+a^2) -20)^2 +b^2) -10)^2 +c^2) -5)^2 +d^2) -2.5)^2 + (sqrt((sqrt(y^2+t^2) -5)^2 +z^2) -2.5)^2 = 1.25
• (((((I))))((II)))
(sqrt((sqrt((sqrt((sqrt(x^2+a^2) -20)^2 +b^2) -10)^2 +c^2) -5)^2 +d^2) -2.5)^2 + (sqrt((sqrt(y^2+z^2) -5)^2 +t^2) -2.5)^2 = 1.25
• (((((I)))I)((I)))
(sqrt((sqrt((sqrt((sqrt(x^2+a^2) -20)^2 +b^2) -10)^2 +c^2) -5)^2 +y^2) -2.5)^2 + (sqrt((sqrt(z^2+d^2) -5)^2 +t^2) -2.5)^2 = 1.25
• (((((I))I))((I)))
(sqrt((sqrt((sqrt((sqrt(x^2+a^2) -20)^2 +b^2) -10)^2 +y^2) -5)^2 +c^2) -2.5)^2 + (sqrt((sqrt(z^2+d^2) -5)^2 +t^2) -2.5)^2 = 1.25
• (((((I)I)))((I)))
(sqrt((sqrt((sqrt((sqrt(x^2+a^2) -20)^2 +y^2) -10)^2 +b^2) -5)^2 +c^2) -2.5)^2 + (sqrt((sqrt(z^2+d^2) -5)^2 +t^2) -2.5)^2 = 1.25
• (((((II))))((I)))
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -20)^2 +a^2) -10)^2 +b^2) -5)^2 +c^2) -2.5)^2 + (sqrt((sqrt(z^2+d^2) -5)^2 +t^2) -2.5)^2 = 1.25
----
• (((((Ia)b)c)d)((It)Y)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*cos(a))^2)-20)^2+(y*cos(b))^2)-10)^2+(y*cos(c))^2)-5)^2+(y*cos(d))^2)-2.5)^2+(sqrt((sqrt(z^2+(y*cos(t))^2)-5)^2(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2)-2.5)^2 = 1.25

• (((((Ia)b)c)d)((IY)t)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*cos(a))^2)-20)^2+(y*cos(b))^2)-10)^2+(y*cos(c))^2)-5)^2+(y*cos(d))^2)-2.5)^2+(sqrt((sqrt((y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2+z^2)-5)^2+(y*cos(t))^2)-2.5)^2 = 1.25

• (((((Ia)b)c)Y)((Id)t)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*cos(a))^2)-20)^2+(y*cos(b))^2)-10)^2+(y*cos(c))^2)-5)^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2)-2.5)^2+(sqrt((sqrt(z^2+(y*cos(d))^2)-5)^2+(y*cos(t))^2)-2.5)^2 = 1.25

• (((((Ia)b)Y)c)((Id)t)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*cos(a))^2)-20)^2+(y*cos(b))^2)-10)^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2)-5)^2+(y*cos(c))^2)-2.5)^2+(sqrt((sqrt(z^2+(y*cos(d))^2)-5)^2+(y*cos(t))^2)-2.5)^2 = 1.25

• (((((Ia)Y)b)c)((Id)t)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*cos(a))^2)-20)^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2)-10)^2+(y*cos(b))^2)-5)^2+(y*cos(c))^2)-2.5)^2+(sqrt((sqrt(z^2+(y*cos(d))^2)-5)^2+(y*cos(t))^2)-2.5)^2 = 1.25

• (((((IY)a)b)c)((Id)t)) - 6 Position Single Start 5-End Rotate, Y:Y->a,b,c,d,t / 0 < a,b,c,d,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2)-20)^2+(y*cos(a))^2)-10)^2+(y*cos(b))^2)-5)^2+(y*cos(c))^2)-2.5)^2+(sqrt((sqrt(z^2+(y*cos(d))^2)-5)^2+(y*cos(t))^2)-2.5)^2 = 1.25
----
• (((((DY)a)b)c)((Id))) - Rotate+Translate X:D(t)->d(t) ; 1-Start,3-End Rotate Y:Y->a,b,c ; -22 < d < 22 / 0 < a,b,c,t < 1.57
(sqrt((sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*((sin(a))*(sin(b))*(sin(c))))^2)-20)^2+(y*cos(a))^2)-10)^2+(y*cos(b))^2)-5)^2+(y*cos(c))^2)-2.5)^2+(sqrt((sqrt(z^2+(x*cos(t)-d*sin(t))^2)-5)^2)-2.5)^2 = 1.25

• (((((DY))a)b)((Id)c)) - Rotate+Translate X:D(t)->d(t) ; 1-Start,3-End Rotate Y:Y->a,b,c ; -22 < d < 22 / 0 < a,b,c,t < 1.57
(sqrt((sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(y*((sin(a))*(sin(b))*(sin(c))))^2)-20)^2+(y*cos(a))^2)-10)^2+(y*cos(b))^2)-5)^2)-2.5)^2+(sqrt((sqrt(z^2+(x*cos(t)-d*sin(t))^2)-5)^2+(y*cos(c))^2)-2.5)^2 = 1.25

• (((((Xd)a)D)c)((I)c)) - Rotate+Translate Y:D(t)->d(t) ; 1-Start,3-End Rotate X:X->a,b,c ; -22 < d < 22 / 0 < a,b,c,t < 1.57
(sqrt((sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))))^2+(y*cos(t)-d*sin(t))^2)-20)^2+(x*cos(a))^2)-10)^2+(y*sin(t)+d*cos(t))^2)-5)^2+(x*cos(b))^2)-2.5)^2+(sqrt((sqrt(z^2)-5)^2+(x*cos(c))^2)-2.5)^2 = 1.25

• (((((Ia)d)b)D)((Z)c)) - Rotate+Translate Y:D(t)->d(t) ; 1-Start,3-End Rotate Z:Z->a,b,c ; -22 < d < 22 / 0 < a,b,c,t < 1.57
(sqrt((sqrt((sqrt((sqrt(x^2+(z*cos(a))^2)-20)^2+(y*cos(t)-d*sin(t))^2)-10)^2+(z*cos(b))^2)-5)^2+(y*sin(t)+d*cos(t))^2)-2.5)^2+(sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))))^2)-5)^2+(z*cos(c))^2)-2.5)^2 = 1.25
----
• (((((Da)b)I)c)((Zd)))
(sqrt((sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(z*cos(a))^2)-20)^2+(z*cos(b))^2)-10)^2+y^2)-5)^2+(z*cos(c))^2)-2.5)^2+(sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))))^2+(x*cos(t)-d*sin(t))^2)-5)^2+0^2)-2.5)^2 = 1.25

• (((((Da)b)I))((Zd)c))
(sqrt((sqrt((sqrt((sqrt((x*sin(t)+d*cos(t))^2+(z*cos(a))^2)-20)^2+(z*cos(b))^2)-10)^2+y^2)-5)^2+0^2)-2.5)^2+(sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))))^2+(x*cos(t)-d*sin(t))^2)-5)^2+(z*cos(c))^2)-2.5)^2 = 1.25








8D (((((II)I)(II))I)(II)) - [Toratiger-Torus x Circle]-tiger , tiger-bundle over the tiger torus , S1 x C2 x S1 x C2 x S1
----------------------------------------------------------------------------------------------------------------------------
(II) - S1
((II)I) - T2
((II)(II)) - S1 x C2
(((II)I)(II)) - S1 x C2 x S1
((((II)(II))I)(II)) - S1 x C2 x S1 x C2
(((((II)I)(II))I)(II)) - S1 x C2 x S1 x C2 x S1

(((((16)8)(8)4)2)(2)1)

(sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -16)^2 + z^2) -8)^2 + (sqrt(w^2 + v^2) -8)^2) -4)^2 + u^2) -2)^2 + (sqrt(t^2 + s^2) -2)^2 = 1

• (((((I))(I)))(I))
(sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -16)^2 + 0^2) -8)^2 + (sqrt(y^2 + 0^2) -8)^2) -4)^2 + 0^2) -2)^2 + (sqrt(z^2 + 0^2) -2)^2 = 1

X,Y -> [xy] : Translate ‘a’ rotate ‘b,c’ /  D -> d : translate ’d’ rotate ’t’

• (((((Xd))(Y))[xy])(D))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(t)-d*sin(t))^2) -16)^2) -8)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -8)^2) -4)^2 + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2)^2 + (sqrt((z*sin(t)+d*cos(t))^2) -2)^2 = 1

• (((((X))(Yd))[xy])(D))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + (z*cos(t)-d*sin(t))^2) -8)^2) -4)^2 + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2)^2 + (sqrt((z*sin(t)+d*cos(t))^2) -2)^2 = 1

• (((((X))(D))[xz])(Zd))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2 + (sqrt((y*sin(t)+d*cos(t))^2) -8)^2) -4)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2)^2 + (sqrt((z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + (y*cos(t)-d*sin(t))^2) -2)^2 = 1
---
• (((((X)[xyz])(Y)))(Z))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -8)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -8)^2) -4)^2) -2)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -2)^2 = 1

• (((((X))(Y))[xyz])(Z))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -8)^2) -4)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -2)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -2)^2 = 1
---
• (((((A)c)(Ca)))(I))
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2 + 0^2) -16)^2 + (y*cos(d)-c*sin(d))^2) -8)^2 + (sqrt((y*sin(d)+c*cos(d))^2 + (x*cos(b)-a*sin(b))^2) -8)^2) -4)^2 + 0^2) -2)^2 + (sqrt(z^2 + 0^2) -2)^2 = 1

• (((((A))(C))c)(Ia))
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2 + 0^2) -16)^2 + 0^2) -8)^2 + (sqrt((y*sin(d)+c*cos(d))^2 + 0^2) -8)^2) -4)^2 + (y*cos(d)-c*sin(d))^2) -2)^2 + (sqrt(z^2 + (x*cos(b)-a*sin(b))^2) -2)^2 = 1
---
• (((((X))(Y)))(I[xy]))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -16)^2 + 0^2) -8)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -8)^2) -4)^2 + 0^2) -2)^2 + (sqrt(z^2 + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2)^2 = 1







8D ((((II)I)((II)I))(II)) - Double Tiger A,B-Duotorus , T3 x C3 x C2 , ((((R1a)R2a)((R1b)R2b)R3)(R1c)r)
-------------------------------------------------------------------------------------------------------
(II) - (1)
((II)I) - ((2.5)1)
((II)(II)) - ((2.5)(2.5)1)
(((II)I)(II)) - (((5)2.5)(2.5)1)
(((II)(II))(II)) - (((5)(5)2.5)(2.5)1)
((((II)I)(II))(II)) - ((((10)5)(5)2.5)(2.5)1)
((((II)I)((II)I))(II)) - ((((10)5)((10)5)2.5)(2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + z^2) -5)^2 + (sqrt((sqrt(w^2 + v^2) -10)^2 + u^2) -5)^2) -2.5)^2 + (sqrt(t^2 + s^2) -2.5)^2 = 1

XYZbox = ±25

• ((((I))((I)))(I))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt((sqrt(y^2 + 0^2) -10)^2 + 0^2) -5)^2) -2.5)^2 + (sqrt(z^2 + 0^2) -2.5)^2 = 1

• ((((X)[xyz])((Y)))(Z))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -5)^2 + (sqrt((sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2) -2.5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -2.5)^2 = 1

• ((((X)[xyzT])((Y)))(Z[xyzt]))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -5)^2 + (sqrt((sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2) -2.5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

• ((((X)[xyzT])((Y[xyzt])))(Z))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -5)^2 + (sqrt((sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -10)^2) -5)^2) -2.5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -2.5)^2 = 1










9D (((II)I)((II)I)((II)I)) - Triger Triotorus , S2xC3xC3 = S2x[T2*T2*T2] , (((maj1)med1)((maj2)med2)((maj3)med3)min)
------------------------------------------------------------------------------------------------------------------------------------------------------------
(sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2c)^2 = R3^2
• (((I))((I))((I)))
(sqrt((sqrt(x^2 + a^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(y^2 + b^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.5^2 = 0
• (((A))((Ca)c)((I)))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + 0^2) -2)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((sqrt(z^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1
• (((A)c)((Ia))((c)))
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt(y^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + 0^2) - 2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1








9D (((((II)I)(II))I)((II)I)) - [Toratiger-Torus x Torus]-tiger , T3xC2xC3 = S1xC2xT2xC2xS1
---------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2b)^2 = Rminor^2
• Diameter Adjustment Equation for Trace Array
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) - b)^2 +0^2) - (b/2))^2 = 1
--- a=15
--- b=7
--- c=4
--- d=1.75
• (((((I))(I)))((I))) : 4x2x4x[R1 pair] of 64 Toruses
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1
—— XYZbox = -32 / +32
—— 55 cubes

• (((((I))(I)))((I))) : Exploratory Function Template
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1

• (((((A)c)(Ca)))((I))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2+(y*cos(d)-c*sin(d))^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2)-3.5)^2 = 1
--- -30 < a < 30 ; -20 < c < 20 ; 0 < b,d < 1.57

• (((((A))(C))c)((Ia))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2)-3.5)^2 = 1

((((())(I)))((II))) : [±R1A±R2] Intercepts are 4 places of ((((I)))((II))) - 1x1x8x[R1 pair] of 16 tori

• (((((A))(C))a)((I)c)) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1

• (((((A))(Ca)))((Ic))) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(y*cos(d)-c*sin(d))^2)-7)^2)-3.5)^2 = 1

• (((((A))(C)))((Ia)c)) : Dual Translate+Rotate
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1

• (((((A))(C))c)((a)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt((x*cos(b)-a*sin(b))^2)-7)^2+z^2)-3.5)^2 = 1

• (((((A))(C))a)((c)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C}
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt((y*cos(d)-c*sin(d))^2)-7)^2+z^2)-3.5)^2 = 1

(((((II)I)(II))I)((II)I)) -> (((((R1a)R2a)(R1b)R3)R4)((R1c)R2b)Rm)

(((((20)10)(10)5)2.5)((5)2.5)1) = Revised diameter values, for better symmetry of lower intercepts

• (((((I))(I)))((I))) :
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -20)^2 +0^2) -10)^2 + (sqrt(y^2+0^2) -10)^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt(z^2+0^2) -5)^2 +0^2) -2.5)^2 = 1

• (((((A)a)(Cz))c)((Z))) : Dual Trans/Rot , Single Rot : X -> A,a , Y -> C,c , Z -> t
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -20)^2 +(x*cos(b)-a*sin(b))^2) -10)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(z*cos(t))^2) -10)^2) -5)^2 +(y*cos(d)-c*sin(d))^2) -2.5)^2 + (sqrt((sqrt((z*sin(t))^2+0^2) -5)^2 +0^2) -2.5)^2 = 1

• (((((Az)a)(C))c)((Z))) : Dual Trans/Rot , Single Rot : X -> A,a , Y -> C,c , Z -> t
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+(z*cos(t))^2) -20)^2 +(x*cos(b)-a*sin(b))^2) -10)^2 + (sqrt((y*sin(d)+c*cos(d))^2+0^2) -10)^2) -5)^2 +(y*cos(d)-c*sin(d))^2) -2.5)^2 + (sqrt((sqrt((z*sin(t))^2+0^2) -5)^2 +0^2) -2.5)^2 = 1

• (((((Az)c)(C))a)((Z))) : Dual Trans/Rot , Single Rot : X -> A,a , Y -> C,c , Z -> t
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+(z*cos(t))^2) -20)^2 +(y*cos(d)-c*sin(d))^2) -10)^2 + (sqrt((y*sin(d)+c*cos(d))^2+0^2) -10)^2) -5)^2 +(x*cos(b)-a*sin(b))^2) -2.5)^2 + (sqrt((sqrt((z*sin(t))^2+0^2) -5)^2 +0^2) -2.5)^2 = 1

• (((((Ac)z)(C))a)((Z))) : Dual Trans/Rot , Single Rot : X -> A,a , Y -> C,c , Z -> t
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+(y*cos(d)-c*sin(d))^2) -20)^2 +(z*cos(t))^2) -10)^2 + (sqrt((y*sin(d)+c*cos(d))^2+0^2) -10)^2) -5)^2 +(x*cos(b)-a*sin(b))^2) -2.5)^2 + (sqrt((sqrt((z*sin(t))^2+0^2) -5)^2 +0^2) -2.5)^2 = 1

• (((((A)y)(Y))c)((C)a)) : Dual Trans/Rot , Single Rot : X -> A,a , Y -> t , Z -> C,c
(sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -20)^2 +(y*cos(t))^2) -10)^2 + (sqrt((y*sin(t))^2+0^2) -10)^2) -5)^2 +(z*cos(d)-c*sin(d))^2) -2.5)^2 + (sqrt((sqrt((z*sin(d)+c*cos(d))^2+0^2) -5)^2 +(x*cos(b)-a*sin(b))^2) -2.5)^2 = 1

• (((((I))(I)))((I))) :
(sqrt((sqrt((sqrt((sqrt(x^2+0^2) -20)^2 +0^2) -10)^2 + (sqrt(y^2+0^2) -10)^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt(z^2+0^2) -5)^2 +0^2) -2.5)^2 = 1








9D ((((II)I)((II)I))((II)I)) - Torus Squared , T2xC3xC3 , ((((R1a)R2a)((R1b)R2b)R3)((R1c)R2c)Rm)
---------------------------------------------------------------------------------------------------
((((II)I)((II)I))((II)I))
(((II)I)((II)I))((II)I)
( ( (II) I) ( (II) I)) ( (II) I)
( ( (xy) z) ( (wv) u)) ( (ts) r)
( ( (x+y) +z) + ( (w+v) +u)) + ( (t+s) +r)
( ( (x+y -R1a) +z -R2a) + ( (w+v -R1b) +u -R2b) -R3) + ( (t+s -R1c) +r -R2c) = Rminor
( ( (x+y -R1a)² +z -R2a)² + ( (w+v -R1b)² +u -R2b)² -R3)² + ( (t+s -R1c)² +r -R2c)² = Rminor²
( ( (√(x+y) -R1a)² +z -R2a)² + ( (√(w+v) -R1b)² +u -R2b)² -R3)² + ( (√(t+s) -R1c)² +r -R2c)² = Rminor²
( ( √((√(x+y) -R1a)² +z) -R2a)² + ( √((√(w+v) -R1b)² +u) -R2b)² -R3)² + ( √((√(t+s) -R1c)² +r) -R2c)² = Rminor²
( √((√((√(x+y) -R1a)² +z) -R2a)² + (√((√(w+v) -R1b)² +u) -R2b)²) -R3)² + (√((√(t+s) -R1c)² +r) -R2c)² = Rminor²
(√((√((√(x+y) -R1a)² +z) -R2a)² + (√((√(w+v) -R1b)² +u) -R2b)²) -R3)² + (√((√(t+s) -R1c)² +r) -R2c)² = Rminor²
(√((√((√(x²+y²) -R1a)² +z²) -R2a)² + (√((√(w²+v²) -R1b)² +u²) -R2b)²) -R3)² + (√((√(t²+s²) -R1c)² +r²) -R2c)² = Rminor²

(sqrt((sqrt((sqrt(x^2+y^2)-R1a)^2+z^2)-R2a)^2+(sqrt((sqrt(w^2+v^2)-R1b)^2+u^2)-R2b)^2)-R3)^2+(sqrt((sqrt(t^2+s^2)-R1c)^2+r^2)-R2c)^2 = Rminor^2
----------------------------
((((12)6)((12)6)3.5)((8)4)1) - Ring-Torus Diameter Values
(sqrt((sqrt((sqrt(x^2+y^2)-12)^2+z^2)-6)^2+(sqrt((sqrt(w^2+v^2)-12)^2+u^2)-6)^2)-3)^2+(sqrt((sqrt(t^2+s^2)-12)^2+r^2)-6)^2 = 1
----------------------------
XYZbox = -30 , +30

• ((((I))((I)))((I))) - 64 tori in 4x4x4 cube array
(sqrt((sqrt((sqrt(x^2+0^2)-12)^2+0^2)-6)^2+(sqrt((sqrt(y^2+0^2)-12)^2+0^2)-6)^2)-3.5)^2+(sqrt((sqrt(z^2+0^2)-8)^2+0^2)-4)^2 = 1

• ((((X)a)((Ib)c))((Id))) -  X:X->a,b,c,d  5-position rotate, single start, 4 end
(sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2)-12)^2+(x*cos(a))^2)-6)^2+(sqrt((sqrt(y^2+(x*cos(b))^2)-12)^2+(x*cos(c))^2)-6)^2)-3.5)^2+(sqrt((sqrt(z^2+(x*cos(d))^2)-8)^2)-4)^2 = 1


• ((((I)a)((Ib)c))((Z)d)) -  Z:Z->a,b,c,d  5-position rotate, single start, 4 end
(sqrt((sqrt((sqrt(x^2)-12)^2+(z*cos(a))^2)-6)^2+(sqrt((sqrt(y^2+(z*cos(b))^2)-12)^2+(z*cos(c))^2)-6)^2)-3.5)^2+(sqrt((sqrt((z*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2)-8)^2+(z*cos(d))^2)-4)^2 = 1









9D ((((II)(II))I)(((II)I)I)) - [Toratiger x Ditorus]-tiger , S1xC2xC2xC3
--------------------------------------------------------------------------

(sqrt((sqrt((sqrt(x^2 + y^2) -R1a)^2 + (sqrt(z^2 + w^2) -R1b)^2) -R2a)^2 + v^2) -R3a)^2 + (sqrt((sqrt((sqrt(u^2 + t^2) -R1c)^2 + s^2) -R2b)^2 + r^2) -R3b)^2 = 1


(II) - (1)
((II)(II)) - ((2.5)(2.5)1)
(((II)I)((II)I)) - (((5)2.5)((5)2.5)1)
((((II)(II))I)((II)I)) - ((((10)(10)5)2.5)((5)2.5)1)
((((II)(II))I)(((II)I)I)) - ((((10)(10)5)2.5)(((10)5)2.5)1)

• ((((II)(II))I)(((II)I)I))
(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + v^2) -2.5)^2 + (sqrt((sqrt((sqrt(u^2 + t^2) -10)^2 + s^2) -5)^2 + r^2) -2.5)^2 = 1

- XYZbox = -25 , +25
- 40 cubes


• ((((I)(I)))(((I)))) : 2x2x8x[R1 pair] of 64 tori
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + 0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2 + 0^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((Az)(Ca))c)(((Z)))) :
(sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2 + (z*cos(t))^2) -10)^2 + (sqrt((y*sin(d)+c*cos(d))^2 + (x*cos(b)-a*sin(b))^2) -10)^2) -5)^2 + (y*cos(d)-c*sin(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(t))^2 + 0^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1


• ((((X)(Y))[xyz])(((Z)))) : Single Trans ‘a’ + Triple Rot ‘b,c,d’ XYZ -> [xyz]
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)(Y)))(((Z)[xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + 0^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)(Y)))(((Z))[xyz]))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + 0^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 + 0^2) -5)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -2.5)^2 = 1

• ((((X)(D))[xz])(((Zd)))) : Single trans ‘a’ + Double Rot ‘b,c’ on X,Z -> [xz] / Trans ‘d’ + Rot ’t’ on Y: D -> d
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(t)+d*cos(t))^2 + 0^2) -10)^2) -5)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + (y*cos(t)-d*sin(t))^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)(D))d)(((Z)[xz]))) 
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(t)+d*cos(t))^2 + 0^2) -10)^2) -5)^2 + (y*cos(t)-d*sin(t))^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)(D)))(((Zd))[xz])) 
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(t)+d*cos(t))^2 + 0^2) -10)^2) -5)^2 + 0^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + (y*cos(t)-d*sin(t))^2) -10)^2 + 0^2) -5)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -2.5)^2 = 1

((((II)(II))I)(((II)I)I))

Dual-Empty 5D intercepts
(((()())I)(((II)I)I)) - Void Intercepts [R1a x R1b]±R2a are 2x2 square of ((((II)I)I)(I)) : 2x Tritoruses ((((II)I)I)I) in 1x1x1x1x2 column
((((II)())I)((()I)I)) - Void Intercepts [R1b x R1c] are 2x2 square of ((((II))I)((I)I)) : 4x Tiger toruses (((II)I)(II)) in 1x1x1x2x1x[R1a pair] row








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The Three Fundamental 9D Toratopes that intercept as 4D arrays of 16 tiger tori (((II)I)(II))

The tesseract array can be passed through cube, square, line, and corner first. The values for b,c,d :

Corner First: b=pi/3 , c=0.9554 , d=pi/4

Line First: b=pi/2 , c=0.9554 , d=pi/4

Square First: b=pi/2 , c=pi/2 , d=pi/4

Cube First: b=pi/2 , c=pi/2 , d=pi/2




• 9D ((((II)(II))(II))((II)I)) - [double tiger x torus]-tiger , ((((I)(I))(I))((I)I)) : 2x2x2x2x1 array of 16 (((II)I)(II)) tiger toruses
-------------------------------------------------------------------------------------------------------------------------------------
3D single-empty intersections of ((((R1a)(R1b)R3)(R1c)R4)((R1d)R2))

((((I)(I))(I))(())) : ring locations at ±R1d±R2 are 4 places along a row of ((((I)(I))(I))), 16x tori ((II)I) in 2x2x2x[Rm pair] cube array
((((I)(I))())((I))) : ring locations at ±R1c are 2 places along a row of ((((I)(I)))((I))), 32x tori ((II)I) in 2x2x4x[R1 pair] tower array
((((I)())(I))((I))) : ring locations at ±R1b are 2 places along a row of ((((I))(I))((I))), 32x tori ((II)I) in 4x2x4 rectangular brick array

(II) - (1)
((II)I) - ((2.5)1)
((II)(II)) - ((2.5)(2.5)1)
(((II)I)(II)) - (((5)2.5)(2.5)1)
((((II)(II))(II))((II)I)) - ((((10)(10)5)(10)2.5)((10)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + (sqrt(v^2 + u^2) -10)^2) -2.5)^2 + (sqrt((sqrt(t^2 + s^2) -10)^2 + r^2) -2.5)^2 = 1

• ((((I)(I))(I))((I)))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + (sqrt(z^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt(w^2 + 0^2) -10)^2 + 0^2) -2.5)^2 = 1

Exploring the 2x2x2x2 tesseract array of 16x Tiger Toruses
• ((((X)(Y))(Z))(([xyz]))) : XYZbox = ±25 , 40 cubes , -28 < a < 28 , 0 < b,c,d < π/2

(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2) -2.5)^2 + (sqrt((sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1







• 9D ((((II)(II))I)((II)(II))) - [toratiger x tiger]-tiger , ((((I)(I))I)((I)(I))) : 2x2x1x2x2 array of 16 (((II)I)(II)) tiger toruses
----------------------------------------------------------------------------------------------------------------------------------
3D single-empty intersections of ((((R1a)(R1b)R2a)R3)((R1c)(R1d)R2b))

((((I)(I)))((I)())) : ring locations at ±R1d are 2 places along a row of ((((I)(I)))((I))), 32x tori ((II)I) in 2x2x4x[R1 pair] tower array
((((I)()))((I)(I))) : ring locations at ±R1b are 2 places along a row of ((((I)))((I)(I))), 32x tori ((II)I) in 2x2x8 tower array

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + v^2) -2.5)^2 + (sqrt((sqrt(u^2 + t^2) -10)^2 + (sqrt(s^2 + r^2) -10)^2) -2.5)^2 = 1

((((10)(10)5)2.5)((10)(10)2.5)1)

• ((((I)(I)))((I)(I)))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + 0^2) -2.5)^2 + (sqrt((sqrt(z^2 + 0^2) -10)^2 + (sqrt(w^2 + 0^2) -10)^2) -2.5)^2 = 1

• ((((X)(Y)))((Z)([xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1









• 9D ((((II)I)(II))((II)(II))) - [tiger torus x tiger]-tiger , ((((I)I)(I))((I)(I))) : 2x1x2x2x2 array of 16 (((II)I)(II)) tiger toruses
---------------------------------------------------------------------------------------------------------------------------------------
3D single-empty intersections of ((((R1a)R2)(R1b)R3a)((R1c)(R1d)R3b))

((((I))(I))((I)())) : ring locations at ±R1d are 2 places along a row of ((((I))(I))((I))), 32x tori in 4x2x4 rectangular brick array
((((I))())((I)(I))) : ring locations at ±R1b are 2 places along a row of ((((I)))((I)(I))), 32x tori in 2x2x8 tower array
(((())(I))((I)(I))) : ring locations at ±R1a±R2 are 4 places along a row of (((I))((I)(I))), 16x tori in 2x2x4 tower array

((((10)5)(10)2.5)((10)(10)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + z^2) -5)^2 + (sqrt(w^2 + v^2) -10)^2) -2.5)^2 + (sqrt((sqrt(u^2 + t^2) -10)^2 + (sqrt(s^2 + r^2) -10)^2) -2.5)^2 = 1

• ((((I))(I))((I)(I)))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt(y^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt(z^2 + 0^2) -10)^2 + (sqrt(w^2 + 0^2) -10)^2) -2.5)^2 = 1

• ((((X))(Y))((Z)([xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1


##########################################################






9D ((((II)(II))((II)I))(II)) : [[Tiger x Torus]-tiger x Circle]-tiger , T2 x C2 x C4
------------------------------------------------------------------------------------------------------
(sqrt((sqrt((sqrt(x^2 + y^2) -R1a)^2 + (sqrt(z^2 + w^2) -R1b)^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1c)^2 + t^2) -R2)^2) -R4)^2 + (sqrt(s^2 + r^2) -R1d)^2 = 1

((((10)(10)5)((10)5)2.5)(10)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + (sqrt((sqrt(v^2 + u^2) -10)^2 + t^2) -5)^2) -2.5)^2 + (sqrt(s^2 + r^2) -10)^2 = 1

• ((((I)(I))((I)))(I))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + (sqrt((sqrt(z^2 + 0^2) -10)^2 + 0^2) -5)^2) -2.5)^2 + (sqrt(w^2 + 0^2) -10)^2 = 1

• ((((X)(Y))((Z)))([xyz]))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 + 0^2) -5)^2) -2.5)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2 + 0^2) -10)^2 = 1

• ((((X)(Y))(([xyz])))(Z))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + (sqrt((sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2 + 0^2) -10)^2 + 0^2) -5)^2) -2.5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 = 1

• ((((X)([xyz]))((Y)))(Z))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2 + 0^2) -10)^2) -5)^2 + (sqrt((sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2 + 0^2) -5)^2) -2.5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2 = 1







9D (((((II)I)I)I)(((II)I)I)) - [tritorus x ditorus]-tiger , S1xC2xC2xC2xS1
-----------------------------------------------------------------------------
Full 3D Intersections
----------------------
(((((II))))(((I)))) - 64x tori ((II)I) in 1x1x8x[R1 octet] column
(((((I)I)))(((I)))) - 64x tori ((II)I) in 2x1x8x[R1 quartet] vertical rectangle array
(((((I))I))(((I)))) - 64x tori ((II)I) in 4x1x8x[R1 pair] vertical rectangle array
(((((I)))I)(((I)))) - 64x tori ((II)I) in 8x1x8 vertical square array
(((((I))))(((II)))) - 64x tori ((II)I) in 1x1x16x[R1 quartet] column
(((((I))))(((I)I))) - 64x tori ((II)I) in 2x1x16x[R1 pair] vertical rectangle array
(((((I))))(((I))I)) - 64x tori ((II)I) in 4x1x16 vertical rectangle array

(((((16)8)4)2)(((8)4)2)1)
(sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -16)^2 + z^2) -8)^2 + w^2) -4)^2 + v^2) -2)^2 + (sqrt((sqrt((sqrt(u^2 + t^2) -8)^2 + s^2) -4)^2 + r^2) -2)^2 = 1

• (((((I)))I)(((I))))
(sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -16)^2 + 0^2) -8)^2 + 0^2) -4)^2 + y^2) -2)^2 + (sqrt((sqrt((sqrt(z^2 + 0^2) -8)^2 + 0^2) -4)^2 + 0^2) -2)^2 = 1

SET: XYZbox = ±37 , 55 cubes

These 8 functions cover all ways to rotate between full 3D intersections

• (((((IY)a)b)c)(((Id)t))) : start Y -> end a,b,c,d,t / 0 < a,b,c,d,t < π/2
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -16)^2 + (y*cos(a))^2) -8)^2 + (y*cos(b))^2) -4)^2 + (y*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt(z^2 + (y*cos(d))^2) -8)^2 + (y*cos(t))^2) -4)^2) -2)^2 = 1

• (((((Ia)Y)b)c)(((Id)t)))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -8)^2 + (y*cos(b))^2) -4)^2 + (y*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt(z^2 + (y*cos(d))^2) -8)^2 + (y*cos(t))^2) -4)^2) -2)^2 = 1

• (((((Ia)b)Y)c)(((Id)t)))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -4)^2 + (y*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt(z^2 + (y*cos(d))^2) -8)^2 + (y*cos(t))^2) -4)^2) -2)^2 = 1

• (((((Ia)b)c)Y)(((Id)t)))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*cos(c))^2) -4)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -2)^2 + (sqrt((sqrt((sqrt(z^2 + (y*cos(d))^2) -8)^2 + (y*cos(t))^2) -4)^2) -2)^2 = 1

• (((((Ia)b)c)d)(((YI)t)))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*cos(c))^2) -4)^2 + (y*cos(d))^2) -2)^2 + (sqrt((sqrt((sqrt((y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2 + z^2) -8)^2 + (y*cos(t))^2) -4)^2) -2)^2 = 1

• (((((Ia)b)c)d)(((It)Y)))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*cos(c))^2) -4)^2 + (y*cos(d))^2) -2)^2 + (sqrt((sqrt((sqrt((y*cos(t))^2 + z^2) -8)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -4)^2) -2)^2 = 1

• (((((Ia)b)c)d)(((It))Y))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*cos(c))^2) -4)^2 + (y*cos(d))^2) -2)^2 + (sqrt((sqrt((sqrt((y*cos(t))^2 + z^2) -8)^2) -4)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -2)^2 = 1

• (((((Ia)b)c)d)(((I)t)Y))
(sqrt((sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) -16)^2 + (y*cos(b))^2) -8)^2 + (y*cos(c))^2) -4)^2 + (y*cos(d))^2) -2)^2 + (sqrt((sqrt((sqrt(z^2) -8)^2 + (y*cos(t))^2) -4)^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))*(sin(t))))^2) -2)^2 = 1

----

These 7 functions will translate and rotate the translating direction on 3 axes combined

• (((((X)[xyz]))Y)(((Z)))) : Single Translate + 3-Axis Rotate X,Y,Z -> [xyz] , -32 < a < 32 , 0 < b,c,d < π/2
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -8)^2) -4)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -8)^2) -4)^2) -2)^2 = 1

• (((((X))[xyz])Y)(((Z))))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -4)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -8)^2) -4)^2) -2)^2 = 1

• (((((X)))Y)(((Z)[xyz])))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2) -4)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -8)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -4)^2) -2)^2 = 1

• (((((X)[xyz])Y))(((Z))))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -16)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -8)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -4)^2 + 0^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -8)^2 + 0^2) -4)^2 + 0^2) -2)^2 = 1

• (((((X))Y)[xyz])(((Z))))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -16)^2 + 0^2) -8)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -4)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -8)^2 + 0^2) -4)^2 + 0^2) -2)^2 = 1

• (((((X))Y))(((Z)[xyz])))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -16)^2 + 0^2) -8)^2 + (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -4)^2 + 0^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -8)^2 + (z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -4)^2 + 0^2) -2)^2 = 1

----

• (((((Xy)))Y)(((Z)[xz]))) : X,Z -> [xz] / Y -> y
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(t) - d*sin(t))^2) -16)^2 + 0^2) -8)^2 + 0^2) -4)^2 + (y*sin(t) + d*cos(t))^2) -2)^2 + (sqrt((sqrt((sqrt((z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -8)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -4)^2 + 0^2) -2)^2 = 1







####################################

6D Toratopes in 4D arrays
---------------------------------------------------------------------

----------16x Duotorus Tigers (((II)I)((II)I)) in 2x2x2x2 tesseract array-----------


• 10D ((((II)(II))(II))(((II)I)I)) - [Double Tiger x Ditorus]-tiger , S1 x C2 x C2 x C4
-----------------------------------------------------------------------------------------
((((10)(10)5)(10)2.5)(((10)5)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + (sqrt(v^2 + u^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt(t^2 + s^2) -10)^2 + r^2) -5)^2 + q^2) -2.5)^2 = 1

• ((((I)(I))(I))(((I))))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + (sqrt(z^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt(w^2 + 0^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1

• ((((X)(Y))(Z))((([xyz]))))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2 + 0^2) -10)^2) -5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2 + 0^2) -10)^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1



• 10D ((((II)(II))(II))((II)(II))) - [Double Tiger x Tiger]-tiger , S1 x C2 x S1 x C5
-----------------------------------------------------------------------------------------
The Full-Intersection trace array ((((I)(I))(I))((I)(I))) produces 32 Tiger Toruses (((II)I)(II)) in 2x2x2x2x2 five dimensional penteract array. Only one Half-Intersection tesseract array at a time is explorable with the X,Y,Z -> [xyz] function. But, using the 5th parameter ’t’ , one can slide along the 5th axis, to alternate between the two tesseract arrays.

There are 3 distinct types of Half-Intersection tesseract arrays in 4D:

((((I)(I))(I))((I)())) - 2 ring locations at ±R1e is a row of ((((I)(I))(I))((I))) , 32x Ditoruses (((II)I)I) in 2x2x2x4 tower
((((I)(I))())((I)(I))) - 2 ring locations at ±R1c is a row of ((((I)(I)))((I)(I))) , 32x Tigers ((II)(II)) in 2x2x2x2x[R1a pair] tesseract
((((I)())(I))((I)(I))) - 2 ring locations at ±R1b is a row of ((((I))(I))((I)(I))) , 32x Tigers ((II)(II)) in 4x2x2x2 brick

Which further slices into 5 types of Quarter-Intersection cube arrays in 3D. There are four copies of these cube arrays, spaced out in a square arrangement. The length and width of this square are two extra higher dimensions, 4D and 5D. Moving by ± both diameter values (R1a, R1b, etc) will place the 3-plane to intersect one of the four cube arrays. A 3D solution of this degree-512 equation can factor out into a product of 128 tori, each having a degree-4 surface. Out of all 128 possible torus intercepts, we can only see 16 or 32 of them at a time, with the 3-plane.

((((I)(I))(I))(()())) - 16 ring locations at (±R1d x ±R1e) ±R2b is 4x4 square of ((((I)(I))(I))) , 16x tori in 2x2x2x[Rm pair] cube array
((((I)(I))())((I)())) - 4 ring locations at ±R1c x ±R1e is 2x2 square of ((((I)(I)))((I))) , 32x tori ((II)I) in 2x2x4x[R1 pair] tower array
((((I)())(I))((I)())) - 4 ring locations at ±R1b x ±R1e is 2x2 square of ((((I))(I))((I))) , 32x tori in 4x2x4 brick array
((((I)())())((I)(I))) - 4 ring locations at ±R1b x ±R1c is 2x2 square of ((((I)))((I)(I))) , 32x tori in 2x2x8 tower array
(((()())(I))((I)(I))) - 16 ring locations at (±R1a x ±R1b) ±R2a is 4x4 square of (((I))((I)(I))) , 16x tori in 2x2x4 tower array

((((R1a)(R1b)R2a)(R1c)R3)((R1d)(R1e)R2b)r)

((((10)(10)5)(10)2.5)((10)(10)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + (sqrt(z^2 + w^2) -10)^2) -5)^2 + (sqrt(v^2 + u^2) -10)^2) -2.5)^2 + (sqrt((sqrt(t^2 + s^2) -10)^2 + (sqrt(r^2 + q^2) -10)^2) -2.5)^2 = 1

• ((((I)(I))(I))((I)(I)))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + (sqrt(y^2 + 0^2) -10)^2) -5)^2 + (sqrt(z^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt(w^2 + 0^2) -10)^2 + (sqrt(v^2 + 0^2) -10)^2) -2.5)^2 = 1

Explore Functions for the 5D penteract array of 32 tiger tori:
---------------------------------------------------------------
Half-Intersection Tesseract arrays at t = ±10 (empty) , Quarter-Intersection Cube arrays at a = ±10 and t = ±10

XYZbox = ±32

X,Y,Z -> [xyz] : translate ‘a’ across 4-space / rotate ‘b,c,d’ / translate ’t’ across 5-space

• ((((X)(Y))(Z))(([xyz])(t)))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2) -2.5)^2 + (sqrt((sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2 + (sqrt(t^2) -10)^2) -2.5)^2 = 1

• ((((X)(Y))(t))((Z)([xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -5)^2 + (sqrt(t^2) -10)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1

• ((((X)(t))(Y))((Z)([xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 + (sqrt(t^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1



• 10D ((((II)I)(II))(((II)I)(II))) - Duo(Tiger Torus) Tiger
-----------------------------------------------------------------------------------------

((((10)5)(10)2.5)(((10)5)(10)2.5)1)

(sqrt((sqrt((sqrt(x^2 + y^2) -10)^2 + z^2) -5)^2 + (sqrt(w^2 + v^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt(u^2 + t^2) -10)^2 + s^2) -5)^2 + (sqrt(r^2 + q^2) -10)^2) -2.5)^2 = 1

• ((((I))(I))(((I))(I)))
(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt(y^2 + 0^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt(w^2 + 0^2) -10)^2) -2.5)^2 = 1

• ((((X))(Y))(((Z))([xyz])))
(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -10)^2) -2.5)^2 + (sqrt((sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -10)^2) -5)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -10)^2) -2.5)^2 = 1







----------16x Toratiger Toruses ((((II)I)(II))I) in 2x2x2x2 tesseract array-----------

• 10D (((((II)(II))(II))((II)I))I) , Tora-[DoubleTiger x Torus]-tiger , T2 x C2 x S1 x C4
---------------------------------------------------------------------------------------------
Tertiary (R3) sweeping of the Tritorus (((II)I)i)I) over the Double Tiger (((II)(II))(II)) : T4 x T2 x C3
Secondary-A (R2a) sweeping of the Toratiger Duotorus ((((II)i)((II)I))I) over the Tiger ((II)(II)) : T2 x C2 x C2 x S1 x C2

(((((I)(I))(I))((I)))I) - 32 Tritoruses ((((II)I)I)I) in 2x2x2x4x1 brick array
----
(((((I)(I))(I))((I)))) - 64 Ditoruses (((II)I)I) in 2x2x2x4x[Rm pair] tower array
(((((I)(I))(I))(()))I) - 4 Locations ±R1d ±R2b are row of (((((I)(I))(I)))I), 32 ditoruses in 2x2x2x1x[R2 pair] cube array
(((((I)(I))())((I)))I) - 2 Locations ±R1c are row of (((((I)(I)))((I)))I), 32 ditoruses in 2x2x4x1x[R1 pair] brick array
(((((I)())(I))((I)))I) - 2 Locations ±R1b are row of (((((I))(I))((I)))I), 32 ditoruses in 4x2x4x1 brick array
----
(((((I)(I))(I))(()))) - 4 Locations ±R1d ±R2b are row of (((((I)(I))(I)))), 32 tori in 2x2x2x[Rm quartet] cube array
(((((I)(I))())((I)))) - 2 Locations ±R1c are row of (((((I)(I)))((I)))), 64 tori in 2x2x4x[R1 pair]x[Rm pair] tower array
(((((I)(I))())(()))I) - 8 Locations ±R1c x (±R1d ±R2b) are 2x4 rectangle of (((((I)(I))))I), 16 tori in 2x2x1x[R1 quartet] square
((((()(I))(I))((I)))) - 2 Locations ±R1a are row of (((((I))(I))((I)))), 64 tori in 4x2x4x[Rm Pair] brick array
((((()(I))(I))(()))I) - 8 Locations ±R1a x (±R1d ±R2b) are 2x4 rectangle of (((((I))(I))I), 8 tori in 4x2x1 flat rectangle array
((((()(I))())((I)))I) - 4 Locations ±R1a x ±R1c are 2x2 square of (((((I)))((I)))I), 32 tori in 8x4x1 flat rectangle array
((((()())(I))((I)))I) - 16 Locations (±R1a x ±R1b) ±R2a are 4x4 square of ((((I))((I)))I), 16 tori in 4x4x1 flat square array

(((((II)(II))(II))((II)I))I)

(((((R1a)(R1b)R2a)(R1c)R3)((R1d)R2b)R4)r)

(((((20)(20)10)(20)5)((20)5)2.5)1)

XYZbox = ±48
60 cubes resolution

(sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -20)^2 + (sqrt(z^2 + w^2) -20)^2) -10)^2 + (sqrt(v^2 + u^2) -20)^2) -5)^2 + (sqrt((sqrt(t^2 + s^2) -20)^2 + r^2) -5)^2) -2.5)^2 + q^2 = 1

• (((((I)(I))(I))((I)))I) 
(sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -20)^2 + (sqrt(y^2 + 0^2) -20)^2) -10)^2 + (sqrt(z^2 + 0^2) -20)^2) -5)^2 + (sqrt((sqrt(w^2 + 0^2) -20)^2 + 0^2) -5)^2) -2.5)^2 + v^2 = 1

X,Y -> [xy] : Translate ‘a’ rotate ‘b,c’ /  D -> d : translate ’d’ rotate ’t’

-40 < a,d < 40
0 < b,c,t < π/2

• (((((X)(Y))(D))(([xy])))d)
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -20)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -20)^2) -10)^2 + (sqrt((z*sin(t) + d*cos(t))^2) -20)^2) -5)^2 + (sqrt((sqrt((y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2) -20)^2) -5)^2) -2.5)^2 + (z*cos(t) - d*sin(t))^2 = 1

• (((((X)(Y))(D))(([xy]d))))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -20)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -20)^2) -10)^2 + (sqrt((z*sin(t) + d*cos(t))^2) -20)^2) -5)^2 + (sqrt((sqrt((y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2 + (z*cos(t) - d*sin(t))^2) -20)^2) -5)^2) -2.5)^2 = 1

• (((((a)(b))(I))((I)))I)  : translate a=±20 ±10 and b=±20 to explore ((((()())(I))((I)))I), analog of ((()())I) cut
(sqrt((sqrt((sqrt((sqrt(a^2) -20)^2 + (sqrt(b^2) -20)^2) -10)^2 + (sqrt(x^2) -20)^2) -5)^2 + (sqrt((sqrt(y^2) -20)^2) -5)^2) -2.5)^2 + z^2 = 1









• 10D (((((II)(II))(II))(II))(II)) - Quadruple Tiger , T4 x C5
-----------------------------------------------------------------
4-Torus embedded into a Clifford 5-Torus

(((((I)(I))(I))(I))(I)) - 32 Tritoruses in 2x2x2x2x2 penteract array
---
((((()(I))(I))(I))(I)) - 2 Locations ±R1a are row of (((((I))(I))(I))(I)), 32 Ditoruses in 4x2x2x2 brick array
(((((I)(I))())(I))(I)) - 2 Locations ±R1c are row of (((((I)(I)))(I))(I)), 32 Ditoruses in 2x2x2x2x[R1 pair] tesseract array
(((((I)(I))(I))())(I)) - 2 Locations ±R1d are row of (((((I)(I))(I)))(I)), 32 Ditoruses in 2x2x2x2x[R2 pair] tesseract array
(((((I)(I))(I))(I))()) - 2 Locations ±R1e are row of (((((I)(I))(I))(I))), 32 Ditoruses in 2x2x2x2x[Rm pair] tesseract array
---
((((()())(I))(I))(I)) - 16 Locations (±R1a x ±R1b) ±R2 are 4x4 square of ((((I))(I))(I)), 16 Tori in 4x2x2 brick array
((((()(I))())(I))(I)) - 4 Locations (±R1a x ±R1c) are 2x2 square of (((((I)))(I))(I)), 32 Tori in 8x2x2 brick array
((((()(I))(I))())(I)) - 4 Locations (±R1a x ±R1d) are 2x2 square of (((((I))(I)))(I)), 32 Tori in 4x2x2x[R1 pair] brick array
((((()(I))(I))(I))()) - 4 Locations (±R1a x ±R1e) are 2x2 square of (((((I))(I))(I))), 32 Tori in 4x2x2x[Rm pair] brick array
(((((I)(I))())())(I)) - 4 Locations (±R1c x ±R1d) are 2x2 square of (((((I)(I))))(I)), 32 Tori in 2x2x2x[R1 quartet] cube array
(((((I)(I))())(I))()) - 4 Locations (±R1c x ±R1e) are 2x2 square of (((((I)(I)))(I))), 32 Tori in 2x2x2x[R1 pair]x[Rm pair] cube
(((((I)(I))(I))())()) - 4 Locations (±R1d x ±R1e) are 2x2 square of (((((I)(I))(I)))), 32 Tori in 2x2x2x[Rm quartet] cube array


(((((16)(16)8)(16)4)(16)2)(16)1)

(sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -16)^2 + (sqrt(z^2 + w^2) -16)^2) -8)^2 + (sqrt(v^2 + u^2) -16)^2) -4)^2 + (sqrt(t^2 + s^2) -16)^2) -2)^2 + (sqrt(r^2 + q^2) -16)^2 = 1

• (((((I)(I))(I))(I))(I))
(sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -16)^2 + (sqrt(y^2 + 0^2) -16)^2) -8)^2 + (sqrt(z^2 + 0^2) -16)^2) -4)^2 + (sqrt(w^2 + 0^2) -16)^2) -2)^2 + (sqrt(v^2 + 0^2) -16)^2 = 1

• ((((([xyz])(t))(X))(Y))(Z))
(sqrt((sqrt((sqrt((sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -16)^2 + (sqrt(t^2) -16)^2) -8)^2 + (sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -4)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -16)^2) -2)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -16)^2 = 1

• (((((t)(X))([xyz]))(Y))(Z))
(sqrt((sqrt((sqrt((sqrt(t^2) -16)^2 + (sqrt((x*sin(b) + a*cos(b))^2) -16)^2) -8)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -16)^2) -4)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -16)^2) -2)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -16)^2 = 1

• (((((X)(Y))(t))([xyz]))(Z))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -16)^2) -8)^2 + (sqrt(t^2) -16)^2) -4)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -16)^2) -2)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -16)^2 = 1

• (((((X)(Y))(Z))(t))([xyz]))
(sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -16)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -16)^2) -8)^2 + (sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -16)^2) -4)^2 + (sqrt(t^2) -16)^2) -2)^2 + (sqrt((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))^2) -16)^2 = 1


in search of combinatorial objects of finite extent
ICN5D
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