Been exploring a really amazing new shape. I'm surprised I've never thought of this one before. Since I've made a mostly complete tour of 6D, I'm on to 7D now. This has got to be one of the coolest ones, for sure. Only one gif, and no pics so far. Also tried out the new rotate function, with nice results.
((((II)I)I)((II)I))
Dimensional Map of ((((II)I)I)((II)I)) Hyperplane Intercepts7D Hyperplane
((((II)I)I)((II)I)) - 1x Tiger Duotorus Torus
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6D Hyperplane Intercepts : Tiger Duotorus (((II)I)((II)I)) = (((R1a)R2a)((R1b)R2b)R3) // Tiger Ditorus ((((II)I)I)(II)) = ((((R1a)R2)R3)(R1b)R4)
((((I)I)I)((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) in 2x1x1x1x1x1 row
((((II))I)((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) as R1a concentric pair
((((II)I))((II)I)) - 2x Tiger Duotoruses (((II)I)((II)I)) as R2a conc pair
((((II)I)I)((I)I)) - 2x Tiger Ditoruses ((((II)I)I)(II)) in 1x1x1x1x2x1 column
((((II)I)I)((II))) - 2x Tiger Ditoruses ((((II)I)I)(II)) as R1b conc pair
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5D Hyperplane Intercepts : Tigritorus (((II)I)(II)) = (((R1a)R2)(R1b)R3) // Tritorus ((((II)I)I)I) = ((((R1)R2)R3)R4)
(((()I)I)((II)I)) - empty, moving out of hole makes row of 2x Tigritori (((II)I)(II)) split/merge
((((I))I)((II)I)) - 4x Tigritori (((II)I)(II)) in 1x1x1x4x1 column
((((I)I))((II)I)) - 4x Tigritori (((II)I)(II)) as R1b conc pair stacked in 1x1x1x2x1 column
((((I)I)I)((I)I)) - 4x Tigritori (((II)I)(II)) in 2x1x1x2x1 vert square array
((((I)I)I)((II))) - 4x Tigritori (((II)I)(II)) as R1b conc pair stacked in 2x1x1x1x1 row
((((II)))((II)I)) - 4x Tigritori (((II)I)(II)) as R1b conc quartet
((((II))I)((I)I)) - 4x Tigritori (((II)I)(II)) as R1a pair stacked in 1x1x1x2x1 column
((((II))I)((II))) - 4x Tigritori (((II)I)(II)) as R1a pair + R1b pair
((((II)I))((I)I)) - 4x Tigritori (((II)I)(II)) as R2 conc pair stacked in 1x1x1x2x1
((((II)I))((II))) - 4x Tigritori (((II)I)(II)) as R1b pair + R2 pair
((((II)I)I)(()I)) - empty, moving out of hole makes column of 2x Tritoruses ((((II)I)I)I) split/merge
((((II)I)I)((I))) - 4x Tritoruses ((((II)I)I)I) in 1x1x1x1x4 column
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4D Hyperplane Non-Empty Intercepts : Tiger ((II)(II)) = ((R1a)(R1b)R2) // Ditorus (((II)I)I) = (((R1)R2)R3)
((((I)))((II)I)) - 8x Ditoruses (((II)I)I) in 1x1x1x8 column
((((I))I)((I)I)) - 8x Tigers ((II)(II)) in 4x1x2x1 vert rectangle array
((((I))I)((II))) - 8x Tigers ((II)(II)) as R1b conc pair in 4x1x1x1 row
((((I)I))((I)I)) - 8x Tigers ((II)(II)) as R1a conc pair in 2x1x2x1 vert square array
((((I)I))((II))) - 8x Tigers ((II)(II)) as R1a pair + R1b pair in 2x1x1x1 row
((((I)I)I)((I))) - 8x Ditoruses (((II)I)I) in 2x1x1x4 vert rectangle array
((((II)))((I)I)) - 8x Tigers ((II)(II)) as R1a conc quartet in 1x1x2x1 column
((((II)))((II))) - 8x Tigers ((II)(II)) as R1a quartet + R1b pair
((((II))I)((I))) - 8x Ditoruses (((II)I)I) as R1 conc pair in 1x1x1x4 column
((((II)I))((I))) - 8x Ditoruses (((II)I)I) as R2 conc pair in 1x1x1x4 column
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3D Hyperplane Non-Empty Intercepts : Torus ((R1)R2)
((((I)))((I)I)) - 16x Tori ((II)I) in 2x1x8 vert rectangle array
((((I)))((II))) - 16x Tori ((II)I) as R1 conc pair in 1x1x8 column
((((I))I)((I))) - 16x Tori ((II)I) in 4x1x4 vert square array+
((((I)I))((I))) - 16x Tori ((II)I) as R1 conc pair in 2x1x4 vert rectangle array
((((II)))((I))) - 16x Tori ((II)I) as R1 conc quartet in 1x1x4 column
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2D Hyperplane Non-Empty Intercepts : Circle : (R)
((((I)))((I))) - 32x Circles (II) in 8x4 rectangle array
Explore functions and basic midcuts:• ((((II)I)I)((II)I)) --> ((((R1)R2a)R3a)((R2b)R3b)R4) --> R1=4.5 , R2=2 , R3=1 , R4=0.5
(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) - 4.5)^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) - 4.5)^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) - 4.5)^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) - 4.5)^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) - 4.5)^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) - 4.5)^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
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Translate Equations
• ((((Ia)b)c)((Id)I))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + c^2) - 1)^2 + (sqrt((sqrt(y^2 + d^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.5^2 = 0
• ((((Ia)b)c)((II)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + c^2) - 1)^2 + (sqrt((sqrt(y^2 + z^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0
• ((((Ia)b)I)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0
• ((((Ia)I)b)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + a^2) - 4)^2 + y^2) - 2)^2 + b^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0
• ((((II)a)b)((Ic)d))
(sqrt((sqrt((sqrt(x^2 + y^2) - 4)^2 + a^2) - 2)^2 + b^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2.5)^2 + d^2) - 1.25)^2 - 0.5^2 = 0
Translate + Rotate Functions
• ((((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
Multi-Position Rotate Functions
• ((((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) - 4.5)^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) - 4.5)^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) - 4.5)^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) - 4.5)^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
And, voi-la, the one animation I made so far:
![Image](http://i.imgur.com/sRik2ko.gif)