The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby Marek14 » Mon Jun 30, 2014 4:51 am

I'm trying to parse all 10D toratopes as well, but that might take a while as there's 2312 of them :)
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 30, 2014 4:57 am

Uhh, yeah! No kidding! But, I knew you were going to do that, eventually. Looking forward to it :)
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Re: The Tiger Explained

Postby Marek14 » Mon Jun 30, 2014 5:13 am

The main problem is updating all 761 9D toratopes to extra torus versions -- that's boring ;)
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 30, 2014 5:30 am

Oh, I'll bet. They seem to take up the most space, too.

Here's a few functions for ((((II)I)I)(II)) that translate + rotate. The last one was what I set out to achieve, and make a gif later on.

• ((((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 - 0.75^2 = 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 (((II)I)(II)), 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
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Re: The Tiger Explained

Postby Marek14 » Tue Jul 01, 2014 3:44 pm

And here's complete cut analysis (up to 3D) of 7D toratopes:
Attachments
7d.txt
(279.83 KiB) Downloaded 77 times
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Re: The Tiger Explained

Postby ICN5D » Tue Jul 01, 2014 6:49 pm

Nice! Funny how we sometimes work on identical things :) I've been developing on a nice, easy to read format for cross section tables. I haven't parsed ALL 7D like you have, but here's one I detailed last night. I'm thinking of taking a look at it's function today, and make photos and videos.



((((II)I)(II)I)I) - 7D Torispheric Tigritorus [21-torus 21-tiger 1-torus]
-------------------------------------------------------------------------------------
INFLATION SEQUENCE : circle -> sphere -> duoring -> circle
----------------------------
DIAMETER HIERARCHY : ((((maj)sec1)(sec2)tert)min)
----------------------------
MINOR - circle
TERTIARY - sphere
SECONDARY 1+2 - duoring
MAJOR - circle
----------------------------
HYPERPLANE INTERCEPT MAP
7D
((((II)I)(II)I)I) - 1x [21-torus 21-tiger 1-torus]
-----------------------------------------------------------------------------------------------------
6D
((((I)I)(II)I)I) - 2x [221-tiger 1-torus] (((II)(II)I)I) stacked 2 along a line in major1 dim
((((II))(II)I)I) - 2x [221-tiger 1-torus] (((II)(II)I)I) as concentric major1 pair
((((II)I)(I)I)I) - 2x [2121-tritorus] ((((II)I)II)I) stacked in 1x1x1x2x1x1 tertiary column
((((II)I)(II))I) - 1x [21-torus 20-tiger 1-torus] ((((II)I)(II))I)
((((II)I)(II)I)) - 2x [21-torus 21-tiger] (((II)I)(II)I) as concentric minor pair
-----------------------------------------------------------------------------------------------------
5D
(((()I)(II)I)I) - empty
((((I))(II)I)I) - 4x [221-ditorus] (((II)II)I) stacked in 1x1x4x1x1 medium column
((((I)I)(I)I)I) - 4x [221-ditorus] (((II)II)I) stacked in 2x1x2x1x1 major/med square array
((((I)I)(II))I) - 2x [220-tiger 1-torus] (((II)(II))I) stacked 2 along a line in major1 dim
((((I)I)(II)I)) - 4x [221-tiger] ((II)(II)I) as concentric minor pair stacked 2 along a line in major1 dim
((((II))(I)I)I) - 4x [221-ditorus] (((II)II)I) as concentric major pair stacked in 1x1x2x1x1 med column
((((II))(II))I) - 2x [220-tiger 1-torus] (((II)(II))I) as concentric maj1 pair
((((II))(II)I)) - 4x [221-tiger] ((II)(II)I) as concentric maj1/minor pairs
((((II)I)()I)I) - empty
((((II)I)(I))I) - 2x tritoruses ((((II)I)I)I) stacked in 1x1x1x2x1 tertiary column
((((II)I)(I)I)) - 4x [212-ditorus] (((II)I)II) as concentric minor pair stacked in 1x1x1x2x1 minor column
((((II)I)(II))) - 2x [21-torus 20-tiger] (((II)I)(II)) as concentric minor pair
-------------------------------------------------------------------------------------------------------------------------
4D non-empty
((((I))(I)I)I) - 8x torispheres ((III)I) stacked in 4x2 major rectangle array
((((I))(II))I) - 4x ditoruses (((II)I)I) stacked in 1x1x4x1 medium column
((((I))(II)I)) - 8x spheritoruses ((II)II) as concentric minor pairs stacked in 1x1x4x1 minor column
((((I)I)(I))I) - 4x ditoruses (((II)I)I) stacked in 2x1x2x1 maj/med square array
((((I)I)(I)I)) - 8x spheritoruses ((II)II) as concentric major/min pairs stacked in 2x1x2x1 maj/minor square array
((((I)I)(II))) - 4x tigers ((II)(II)) as concentric minor pairs stacked 2 along line in major1 dim
((((II))(I))I) - 4x ditoruses (((II)I)I) as concentric major pair stacked in 1x1x2x1 med column
((((II))(I)I)) - 8x spheritoruses ((II)II) as concentric major/min pairs stacked in 1x1x2x1 minor column
((((II))(II))) - 4x tigers ((II)(II)) as concentric maj1/minor pairs
((((II)I)(I))) - 4x ditoruses (((II)I)I) as concentric minor pair stacked in 1x1x1x2 minor column
--------------------------------------------------------------------------------------------------------------------------
3D non-empty
((((I))(I))I) - 8x torii ((II)I) stacked in 4x2 major rectangle array
((((I))(I)I)) - 16x spheres (III) as concentric pair stacked in 4x2 rectangle array
((((I))(II))) - 8x torii ((II)I) as concentric minor pair stacked in 1x1x4 minor column
((((I)I)(I))) - 8x torii ((II)I) as concentric minor pair stacked in 2x1x2 maj/min square array
((((II))(I))) - 8x torii ((II)I) as concentric major/minor pairs stacked in 1x1x2 minor column
--------------------------------------------------------------------------------------------------------------------
2D non-empty
((((I))(I))) - 32x circles (II) as concentric pair stacked in 4x2 rectangle array


Construction Flowchart for [21-torus 21-tiger 1-torus]
--------------------------------------------------------------------
Code: Select all
2D      3D          4D            5D               6D                 7D                   
                     
                              (((II)I)II)                                     

                 ((III)I)     (((II)II)I)     (((II)(II)I)I)
                                             
      ((II)I)   ((II)(II))   (((II)I)(II))   ((((II)I)(II))I)                           
(II)                                                            ((((II)I)(II)I)I)
       (III)     ((II)II)    (((II)(II))I)    (((II)I)(II)I)           
                             
                (((II)I)I)    ((II)(II)I)     ((((II)I)II)I)                                 
                                     
                             ((((II)I)I)I)             
Last edited by ICN5D on Wed Aug 06, 2014 5:00 am, edited 1 time in total.
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Re: The Tiger Explained

Postby Marek14 » Tue Jul 01, 2014 9:35 pm

I see you have a bit different definition of major dimensions than me. I count "major" by number of inner parenthesis, you count by number of outer parenthesis. The 21-torus 21-tiger 1-torus wold be ((((maj1)sec)(maj2)tert)min) in my system. The contentious pair of parentheses is on third level from outside (which is what you count), but on first level from inside (which is what I count).
So, for me, "major" dimensions are those that can be freely increased. Any subordinate dimension cannot be increased freely since the shape would eventually become self-intersecting, but the major dimensions can be arbitrarily large.

I think this is caused by our differing approaches: you see the toratopes from their minor elements (minor element along a major element) while I see them from major elements (major element inflated by minor element). Both are correct, but if you'll read my 7D file, you might get confused by me marking some dimensions and diameters as major while you wouldn't do so.

I also think that my description for cuts is a bit more compact than yours, mainly because of previous definition. For example, there is no need specifying "concentric" minor pair -- in my system, ALL pairs are concentric.

A quick review how it works: <number> of <toratope> signifies that the toratopes are arranged in a single line along one of their major dimensions. If there are multiple kinds of major dimensions, this is signified by brackets (so there are two[torus] torus tigers ((((I)I)I)(II)), a cut of ditorus tiger, and two[circle] torus tigers (((II)I)((I)I))), a cut of duotorus tiger).
Arrays are arranged along multiple major dimensions. There can be variants, for example a 2x2 array of tigers can be A/A (((I)(I))(II)), a cut of double tiger, or A/B (((I)I)((I)I)), a cut of duotorus tiger.
Arrangements in non-major dimensions are "stacks", which I still call "vertical" for special case of toruses, otherwise with the dimensional marking.

Now the question is how to classify the empty cuts. I came out with the following system:

() is called "eye".
(()I) is "two points eye" or "dyad eye". If you evolve this shape in the empty dimension into ((I)I) (so the eye gets a slit pupil inside), you will see the dyad in the maximum points.
(()II) is "circle eye", (()III) "sphere eye", etc. The shape has two meanings - it's what will result when we remove the eye, and also what will be the maximum cut when we evolve the cut in the eye's direction.

Now let's look at torus eye.

Torus is ((II)I) and an eye can be inserted in two points. You have a "major torus eye" ((()II)I) and a "minor torus eye" ((II)()I).
But configurations like "two circles" would be considered of torus species, so there is also "major two circles eye" ((()I)I) and "minor two circles eye" ((I)()I).

Multiple eyes result in things like "major torus bi-eye" ((()()II)I), "major/minor torus bi-eye" ((()II)()I) and "minor torus bi-eye" ((II)()()I).

Nested eyes are called by their species name, so a "circle eye-torus" is ((())II) since (()) is torus species and a "major tiger eye-tiger" would be (((()())II)(II)), an empty 4D slice of triple tiger.
If you have more than one kind of eye, the nomenclature would look like "torus with major eye and minor eye-torus" ((()II)(())I).

This would allow more precise naming of empty cuts than just "Empty cut whatever".
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Re: The Tiger Explained

Postby ICN5D » Wed Jul 02, 2014 1:18 am

I think this is caused by our differing approaches: you see the toratopes from their minor elements (minor element along a major element) while I see them from major elements (major element inflated by minor element). Both are correct, but if you'll read my 7D file, you might get confused by me marking some dimensions and diameters as major while you wouldn't do so.


Yeah, I noticed that. It's a new thing I've been trying out lately. Starting in 5D, we have arbitrary inflation sequences with the same outcome. This is where I searched for a systematic construction method. The way I reduce it is from reading right to left, for the inflation sequence. So, say ((((II)I)(II)I)I) for example again:

Originally,

circle -> sphere -> duoring -> circle

(II) - circle
(!I) - along sphere
((III)I) - torisphere
((!!I)I) - along duoring
(((II)(II)I)I) - 221-tiger 1-torus
(((!I)(II)I)I) - along circle
((((II)I)(II)I)I) - 21-torus 21-tiger 1-torus

giving ((((maj)sec1)(sec2)tert)min)



And the way you reduce it is by innermost brackets, in this case a swap in the order. Which actually follows a more consistent path of torisphere -> tiger :


circle -> sphere -> circle -> duoring

(II) - circle
(!I) - along sphere
((III)I) - torisphere
((!II)I) - along circle
(((II)II)I) - 221-ditorus
(((!I)!I)I) - along duoring
((((II)I)(II)I)I) - 21-torus 21-tiger 1-torus

producing ((((maj1)sec)(maj2)tert)min)

where the diameters are shaped like:

MAJ 1,2 - duoring
SEC - circle
TERT - sphere
MIN - circle


In which I'm inclined to agree, based on :

So, for me, "major" dimensions are those that can be freely increased. Any subordinate dimension cannot be increased freely since the shape would eventually become self-intersecting, but the major dimensions can be arbitrarily large.



By looking at the 4x2 array of concentric circles ((((I))(I))) , we have 2 free directions to increase the 2 equal major diameters. According to ((((maj1)sec)(maj2)tert)min) , the secondary diameter is what I considered to be part the duoring, but will lead to self-intersecting if too large.


So, how about ((((II)(II))I)(II)) , tiger -> tiger , or in this case ditorus -> trioring :

circle -> circle -> circle -> trioring

(II) - circle
(!I) - along circle
((II)I) - torus
((!I)I) - along circle
(((II)I)I) - ditorus
(((!!)I)!) - along trioring
((((II)(II))I)(II)) - 220-tiger 1-torus 20-tiger

giving ((((maj1)(maj2)sec)tert)(maj3)min) , according to inflating a large shape trioring with a small shape ditorus.

This produces a cube array of major pairs of 16 torii, allowing three free dimensions to increase, and thus three equal major diameters made by inflating a trioring. The secondary, tertiary, and minor come from the original ditorus, and after inflation, stay in their place on the totem pole of diameter ranking.


This makes me want to try something new, animated diameter values! And, showing them in the axial cut pictures, too. That would be neat. All separations, be it pairing or stacking, are diameters cut open, revealing their hollow structure inside.
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Re: The Tiger Explained

Postby Marek14 » Wed Jul 02, 2014 5:31 am

Sounds fun :)
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Re: The Tiger Explained

Postby ICN5D » Wed Jul 02, 2014 6:03 am

All right, so what's the deal, here? I've google searched just about all forms of " multidimensional torus" or "toroid cross section", but I can't find anything like our discussions. No references to A000669, or the implicit functions, or multitangent Villarceau sections. What little there is stops at 4D, like a few youtube vids of the ditorus cut of displaced torii or a computer microchip. There's nothing even close to what this thread turned into, even in the mathematical PDF papers. And, there's a lot of them. Is this some sort of unexplored field of research? Are these GIFs and pics I make something of a novelty? It's actually kind of encouraging. Something new for the world, right?
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Re: The Tiger Explained

Postby ICN5D » Wed Jul 02, 2014 6:48 am

Here's the functions of ((((II)I)(II)I)I) I made today, along with the exploration script. Each function is labelled by its step in the script:





Root Equation
((((II)I)(II)I)I) = ((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


Base Form Equation
• ((((Ii)i)(Ii)i)I) - 8x torii ((II)I) stacked in 4x2 major rectangle array
((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + c^2) - 2)^2 + d^2 - 1.75)^2 + z^2 - 1^2 = 0

• ((((Ii)i)(Ii)I)i) - 16x spheres (III) as concentric pair stacked in 4x2 rectangle array
((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + c^2) - 2)^2 + z^2 - 1.75)^2 + d^2 - 1^2 = 0

• ((((Ii)i)(II)i)i) - 8x torii ((II)I) as minor pair stacked in 1x1x4 minor column
((sqrt((sqrt(x^2 + a^2) - 4)^2 + b^2) - 2)^2 + (sqrt(y^2 + z^2) - 2)^2 + c^2 - 1.75)^2 + d^2 - 1^2 = 0

• ((((Ii)I)(Ii)i)i) - 8x torii ((II)I) as minor pair stacked in 2x1x2 maj/min square array
((sqrt((sqrt(x^2 + a^2) - 4)^2 + y^2) - 2)^2 + (sqrt(z^2 + b^2) - 2)^2 + c^2 - 1.75)^2 + d^2 - 1^2 = 0

• ((((II)i)(Ii)i)i) - 8x torii ((II)I) as major/minor pairs stacked in 1x1x2 minor column
((sqrt((sqrt(x^2 + y^2) - 4)^2 + a^2) - 2)^2 + (sqrt(z^2 + b^2) - 2)^2 + c^2 - 1.75)^2 + d^2 - 1^2 = 0



Exploration Functions - S5, S6, S7 are combined single rotate & rotate+translate
• S1 ((((Xy)z)(Yc)x)Z) - ((((I))(I))I)
((sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 4)^2 + (z*cos(b))^2) - 2)^2 + (sqrt((y*sin(a))^2 + c^2) - 2)^2 + (x*cos(d))^2 - 1.25)^2 + (z*sin(b))^2 - 1^2 = 0

• S2 ((((Xz)x)(Yi)y)Z) - ((((I))(I))I)
((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4)^2 + (x*cos(b))^2) - 2)^2 + (sqrt((y*sin(d))^2 + c^2) - 2)^2 + (y*cos(d))^2 - 1.25)^2 + (z*sin(a))^2 - 1^2 = 0

• S3 ((((Xz)y)(Yx)Z)i) - ((((I))(I)I))
((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 4)^2 + (y*cos(b))^2) - 2)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2)^2 + (z*sin(a))^2 - 1.75)^2 + d^2 - 1^2 = 0

• S4 ((((Ai)c)(Ca)i)I) - ((((I))(I))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2 - 1.75)^2 + z^2 - 1^2 = 0
-- Range: A,C=-8,+8 / B,D=0,1.57
--- B=1.2 , C=0 : animate A, alternate D [A,D] = [-6,0]>[6,0]>[6,1.57]>[-6,1.57]>[-6,0]
--- A,C=0 , B=1.2 : animate D [0] > [1.57] > [0]

• S5 ((((Az)y)(Ya)i)Z) - ((((I))(I))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d))^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2 - 1.75)^2 + (z*sin(d))^2 - 1^2 = 0
--Range: A=-8,+8 / B,C,D=0,1.57

• S6 ((((Ai)y)(Ya)z)Z) - ((((I))(I))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + (z*cos(d))^2 - 1.75)^2 + (z*sin(d))^2 - 1^2 = 0
--Range: A=-8,+8 / B,C,D=0,1.57
--- A=2 : alternate [B,C] = [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0]

• S7 ((((Ad)y)(Ya)i)I) - ((((I))(I))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + d^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + 0^2 - 1.75)^2 + z^2 - 1^2 = 0
--Range: A,D=-8,+8 / B,C=0,1.57
---A=1.67 / B=1.11 / C=0 : adj D for amazing topology morphs!!



I should have some animations coming for those labelled with addendum notes. Those functions had the most interesting things going on.
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Re: The Tiger Explained

Postby Marek14 » Wed Jul 02, 2014 7:39 pm

ICN5D wrote:All right, so what's the deal, here? I've google searched just about all forms of " multidimensional torus" or "toroid cross section", but I can't find anything like our discussions. No references to A000669, or the implicit functions, or multitangent Villarceau sections. What little there is stops at 4D, like a few youtube vids of the ditorus cut of displaced torii or a computer microchip. There's nothing even close to what this thread turned into, even in the mathematical PDF papers. And, there's a lot of them. Is this some sort of unexplored field of research? Are these GIFs and pics I make something of a novelty? It's actually kind of encouraging. Something new for the world, right?


It's possible that this is an unexplored area. Certainly we have material for at least one paper, possibly several. I suspect we might just be not that interested in those... I myself have an autistic spectrum disorder and dropped out of several colleges because I couldn't handle the administrative, so writing an article and trying to get it accepted somewhere would be very hard for me :)
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Re: The Tiger Explained

Postby ICN5D » Fri Jul 04, 2014 10:26 pm

You know, something I've noticed is that Asperger's has a strong correlation with visualizing very high dimensions. It seems to be almost a requirement, to have the capacity to apply the pattern recognition, and differentiation across dimensional analogy. I have a very narrow focus, that bounces around to many different things. School is tough for me, too. Unless, it's a subject I feel is worth learning. I can only learn the things I want to learn, unfortunately. Trying to forcefully cram information I don't care for just doesn't work. It doesn't surprise me that you're a translator, language grammar is just an abstract math. I've probably invented like 50 different languages based on mathematical algorithms that transform english. So, go figure that when I came across a math problem in high dimension geometry, I created a mathematical language around it.


But, on another note, I did some snooping around Wikipedia today, and found some of the widely accepted terms for the toratopes. Apparently, their construction follows a system called a ' bundle over n-sphere'. A torus is a circle bundle over the circle , T2 = S1xS1. A ditorus is a T3 = S1xS1xS1 . And, a Clifford torus is C2 , which helps big time with the tiger.


(IIII) - S3
((III)I) - S1xS2
((II)II) - S2xS1
(((II)I)I) - S1xS1xS1 = T3
((II)(II)) - S1xC2 = S1x[S1*S1]

Where all make the five distinct 3-manifolds in 4D

This is basically the inflation sequence. Which is great, because now we can provide this notation as well to describe them, and show its relation to toratopic notation. And, come to find out, this algebraic topology is widely studied in depth, but in a very generalized way. Not much goes into the intricacies of 5D+ toratopes, and their unique structures. And certainly none of them are rendered visually. I also came across another neat way to classify them, based on how to fold their n-cube nets. Toratopic notation directly reflects the process, through spheration and the comb product of gluing sides together. For once, it's nice to understand a math article on Wikipedia :D .


So, how about a comparison of bundle over n-sphere terminology with toratopic notation:

For the Clifford tori, I'm using brackets [Sn*Sm] to represent a product of C(n+m), where C2 = [S1*S1] . In the case of a cylspherinder margin, which is a bisecting rotation of C2 , I'll use [S2*S1] , and for margin of a bi-spherical prism , [S2*S2] . A (torus*circle) margin can be written as C2xS1 = [T2*S1], and a (torus*torus) prism margin as [T2*T2] , which also can be written C2xC2 , Clifford torus bundle over Clifford torus. But, as an alternative, I suppose I could represent [S2*S1] as C(2*1) , making a complex Clifford 3-manifold. But, it would fall short when trying to define [T2*S1].


2-Manifolds in 3D:
(III) - S2
((II)I) -T2 = S1xS1


All 5 distinct 3-Manifolds in 4D:
(IIII) - S3
((III)I) - S1xS2
((II)II) - S2xS1
(((II)I)I) - T3 = S1xS1xS1
((II)(II)) - S1xC2 = S1x[S1*S1]


All 12 distinct 4-Manifolds in 5D:
(IIIII) - S4
((II)III) - S3xS1
((II)(II)I) - S2xC2 = S2x[S1*S1]
((III)II) - S2xS2
(((II)I)II) - S2xT2 = S2xS1xS1
((III)(II)) - S1x[S2*S1]
(((II)I)(II)) - S1xC2xS1 = S1x[T2*S1] = S1x[(S1xS1)*S1] = S1x[S1*S1]xS1 , the final S1 from T2 commutes to the outside right
((IIII)I) - S1xS3
(((II)II)I) - S1xS2xS1
(((II)(II))I) - T2xC2 = S1xS1x[S1*S1]
(((III)I)I) - T2xS2 = S1xS1xS2
((((II)I)I)I) - T4 = S1xS1xS1xS1


All 33 distinct 5-manifolds in 6D:
(IIIIII) - S5
((II)IIII) - S4xS1
((II)(II)II) - S3xC2 = S3x[S1*S1]
((II)(II)(II)) - S1xC3 = S1x[S1*S1*S1]
((III)III) - S3xS2
(((II)I)III) - S3xT2 = S3xS1xS1
((III)(II)I) - S2x[S2*S1]
(((II)I)(II)I) - S2xS1xC2 = S2x[T2*S1] = S2x[(S1xS1)*S1] = S2x[S1*S1]xS1 = S2xS1x[S1*S1]
((III)(III)) - S1x[S2*S2]
(((II)I)(III)) - S1x[T2*S2] = S1x[(S1xS1)*S2] = S1x[S2*S1]xS1
(((II)I)((II)I)) - S1xC2xC2 = S1x[T2*T2] = S1x[(S1xS1)*(S1xS1)]
((IIII)II) - S2xS3
(((II)II)II) - S2xS2xS1
(((II)(II))II) - S2xS1xC2 = S2xS1x[S1*S1]
(((III)I)II) - S2xS1sS2
((((II)I)I)II) - S2xT3 = S2xS1xS1xS1
((IIII)(II)) - S1x[S3*S1]
(((II)II)(II)) - S1xS2xC2 = S1xS2x[S1*S1] = S1x[(S2xS1)*S1] = S1x[S2*S1]xS1
(((II)(II))(II)) - T2xC3 = S1xS1x[S1*S1*S1]
(((III)I)(II)) - S1xC2xS2 = S1x[(S1xS2)*S1] = S1x[S1*S1]xS2
((((II)I)I)(II)) - S1xC2xT2 = S1x[T3*S1] = S1x[S1*S1]xT2 = S1x[(S1xS1xS1)*S1] = S1x[S1*S1]xS1xS1
((IIIII)I) - S1xS4
(((II)III)I) - S1xS3xS1
(((II)(II)I)I) - S1xS2xC2 = S1xS2x[S1*S1]
(((III)II)I) - S1xS2xS2
((((II)I)II)I) - S1xS2xT2 = S1xS2xS1xS1
(((III)(II))I) - T2x[S2*S1] = S1xS1x[S2*S1]
((((II)I)(II))I) - T2xC2xS1 = S1xS1x[(S1xS1)*S1] = S1xS1x[S1*S1]xS1
(((IIII)I)I) - T2xS3 = S1xS1xS3
((((II)II)I)I) - T2xS2xS1 = S1xS1xS2xS1
((((II)(II))I)I) - T3xC2 = S1xS1xS1x[S1*S1]
((((III)I)I)I) - T3xS2 = S1xS1xS1xS2
(((((II)I)I)I)I) - T5 = S1xS1xS1xS1xS1



I like this, actually. If more people are watching this thread from a strong topology background, they might understand these better.

EDIT 7/5: Correcting minor errors, (((III)I)I) = T2xS2 = S1xS1xS2
Last edited by ICN5D on Sat Jul 05, 2014 11:27 pm, edited 1 time in total.
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Re: The Tiger Explained

Postby ICN5D » Sat Jul 05, 2014 2:48 am

All 90 distinct 6-Manifolds in 7D:

Sn = n-sphere
Tn = n-torus
Cn = n-Clifford manifold , [Sn*Sn] prism D-2 margin
[Sn*Sm] = C(n+m) complex Clifford manifold, D-2 margin of cylindrical prism
[(SnxSm)*Sk] = complex C(n+m+k) Clifford manifold , D-2 margin of cylindrical prism



Q x R = 'Q' bundle over the 'R'


1. Heptasphere (IIIIIII) - S6
2. 61-torus ((IIIIII)I) - S1xS5
3. 511-ditorus (((IIIII)I)I) - T2xS4
4. 4111-tritorus ((((IIII)I)I)I) - T3xS3
5. 31111-tetratorus (((((III)I)I)I)I) - T4xS2
6. Pentatorus ((((((II)I)I)I)I)I) - T6
7. Tiger tritorus (((((II)(II))I)I)I) - T4xC2
8. 22111-tetratorus (((((II)II)I)I)I) - T3xS2xS1
9. 320-tiger 11-ditorus ((((III)(II))I)I) - T3x[S2*S1]
10. Torus tiger ditorus (((((II)I)(II))I)I) - T3xC2xS1
11. 3211-tritorus ((((III)II)I)I) - T2xS2xS2
12. 21211-tetratorus (((((II)I)II)I)I) - T2xS2xT2
13. 221-tiger 11-ditorus ((((II)(II)I)I)I) - T2xS2xC2
14. 2311-tritorus ((((II)III)I)I) - T2xS3xS1
15. 420-tiger 1-torus (((IIII)(II))I) - T2x[S3*S1]
16. 31-torus 20-tiger 1-torus ((((III)I)(II))I) - T2xC2xS2 = T2x[S1*S1]xS2 = T2x[(S1xS2)*S1]
17. Ditorus tiger torus (((((II)I)I)(II))I) - T2xC2xT2 = T2x[T3*S1] = T2x[S1*S1]xT2
18. Double tiger torus ((((II)(II))(II))I) - T3xC3 = T2xC2xC2 = T2x[(S1xC2)*S1]
19. 22-torus 20-tiger 1-torus ((((II)II)(II))I) - T2x[(S2xS1)*S1] = T2x[S2*S1]xS1
20. 421-ditorus (((IIII)II)I) - S1xS2xS3
21. 3121-tritorus ((((III)I)II)I) - S1xS2xS1xS2
22. 21121-tetratorus (((((II)I)I)II)I) - S1xS2xT3
23. 220-tiger 21-ditorus ((((II)(II))II)I) - S1xS2xS1xC2
24. 2221-tritorus ((((II)II)II)I) - S1xS2xS2xS1
25. 330-tiger 1-torus (((III)(III))I) - T2x[S2*S2]
26. 21-torus 30-tiger 1-torus ((((II)I)(III))I) - T2x[S2*S1]xS1 = T2[T2*S2]
27. Duotorus tiger torus ((((II)I)((II)I))I) - T2xC2xC2
28. 321-tiger 1-torus (((III)(II)I)I) - S1xS2x[S2*S1]
29. 21-torus 21-tiger 1-torus ((((II)I)(II)I)I) - S1xS2xC2xS1 = S1xS2x[T2*S1]
30. 331-ditorus (((III)III)I) = S1xS3xS2
31. 2131-tritorus ((((II)I)III)I) = S1xS3xT2
32. Triger torus (((II)(II)(II))I) = S1xS2xC3
33. 222-tiger 1-torus (((II)(II)II)I) - S1xS3xC2
34. 241-ditorus (((II)IIII)I) - S1xS4xS1
35. 520-tiger ((IIIII)(II)) - S1x[S4*S1]
36. 41-torus 20-tiger (((IIII)I)(II)) - S1xC2xS3 = S1x[(S1xS3)*S1]
37. 311-ditorus 20-tiger ((((III)I)I)(II)) - S1xC2xS1xS2 = S1x[(T2xS2)*S1] = S1x[T2*S1]xS2
38. Tritorus tiger (((((II)I)I)I)(II)) - S1xC2xT3 = S1x[T4*S1]
39. Tiger torus tiger ((((II)(II))I)(II)) - S1xC2xS1xC2
40. 221-ditorus 20-tiger ((((II)II)I)(II)) - S1xC2xS2xS1 = S1x[(S1xS2xS1)*S1] = S1x[(S1xS2)*S1]xS1
41. 320-tiger 20-tiger (((III)(II))(II)) - S1xC2x[S2*S1]
43. 32-torus 20-tiger (((III)II)(II)) - S1x[S2*S1]xS2 = S1x[(S2xS2)*S1]
44. 212-ditorus 20-tiger ((((II)I)II)(II)) - S1x[S2*S1]xT2 = S1x[(S2xT2)*S1]
45. 221-tiger 20-tiger (((II)(II)I)(II)) - S1x[S2*S1]xC2 = S1x[(S2xC2)*S1]
46. 23-torus 20-tiger (((II)III)(II)) - S1x[S3*S1]xS1 = S1x[(S3xS1)*S1]
47. 52-torus ((IIIII)II) - S2xS4
48. 412-ditorus (((IIII)I)II) - S2xS1xS3
49. 3112-tritorus ((((III)I)I)II) - S2xT2xS2
50. 21112-tetratorus (((((II)I)I)I)II) - S2xT3xS1
51. 220-tiger 12-ditorus ((((II)(II))I)II) - S2xT2xC2
52. 2212-tritorus ((((II)II)I)II) - S2xS1xS2xS1
53. 320-tiger 2-torus (((III)(II))II) - S2xS1x[S2*S1]
54. 21-torus 20-tiger 2-torus ((((II)I)(II))II) - S2xS1xC2xS1 = S2xS1x[T2*S1]
55. 322-ditorus (((III)II)II) - S2xS2xS2
56. 2122-tritorus ((((II)I)II)II) - S2xS2xT2
57. 221-tiger 2-torus (((II)(II)I)II) - S2xS2xC2
58. 232-ditorus (((II)III)II) - S2xS3xS1
59. 430-tiger ((IIII)(III)) - S1x[S3*S2]
60. 21-torus 40-tiger (((II)I)(IIII)) - S1x[T2*S3] = S1x[S3*S1]xS1
61. 31-torus 30-tiger (((III)I)(III)) - S1x[(S1xS2)*S2] = S1x[S2*S1]xS2 = S1x[S2*S2]xS1
62. 31-torus 21-torus 0-tiger (((III)I)((II)I)) - S1xC2x[S2*S1] = S1x[(S1xS2)*T2] = S1x[T2*S1]xS2
63. 211-ditorus 30-tiger ((((II)I)I)(III)) - S1x[T3*S2] = S1x[S2*S1]xT2
64. Ditorus/torus tiger ((((II)I)I)((II)I)) - S1xC2xC2xS1 = S1x[T3*T2]
65. 220-tiger 30-tiger (((II)(II))(III)) - S1x[(S1xC2)*S2] = S1x[S2*S1]xC2
66. Tiger/torus tiger (((II)(II))((II)I)) - S1x[(S1xC2)*T2] = S1x[T2*S1]xC2
67. 22-torus 30-tiger (((II)II)(III)) - S1x[(S2xS1)*S2] = S1x[S2*S2]xS1
68. 22-torus 21-torus 0-tiger (((II)II)((II)I)) - S1x[(S2xS1)*T2] = S1x[S2*S1]xC2 = S1x[T2*S2]xS1
69. 421-tiger ((IIII)(II)I) - S2x[S3*S1]
70. 31-torus 21-tiger (((III)I)(II)I) - S2xC2xS2 = S2x[(S1xS2)*S1]
71. 211-ditorus 21-tiger ((((II)I)I)(II)I) - S2xC2xT2 = S2x[T3*S1]
72. 220-tiger 21-tiger (((II)(II))(II)I) - S2xS1xC3 = S2xC2xC2 = S2x[(S1xC2)*S1]
73. 22-torus 21-tiger (((II)II)(II)I) - S2x[(S2xS1)*S1] = S2x[S2*S1]xS1
74. 43-torus ((IIII)III) - S3xS3
75. 313-ditorus (((III)I)III) - S3xS1xS2
76. 2113-tritorus ((((II)I)I)III) - S3xT3
77. 220-tiger 3-torus (((II)(II))III) - S3xS1xC2
78. 223-ditorus (((II)II)III) - S3xS2xS1
79. 331-tiger ((III)(III)I) - S2x[S2*S2]
80. 21-torus 31-tiger (((II)I)(III)I) - S2x[T2*S2] = S2x[S2*S1]xS1
81. 21-torus 21-torus 1-tiger (((II)I)((II)I)I) = S2xC2xC2 = S2x[T2*T2]
82. 3220-triger ((III)(II)(II)) - S2x[S2*S1*S1]
83. Torus triger (((II)I)(II)(II)) - S2xC3xS1 = S2x[T2*S1*S1]
84. 322-tiger ((III)(II)II) - S3x[S2*S1]
85. 21-torus 22-tiger (((II)I)(II)II) - S3xC2xS1 = S3x[T2*S1]
86. 34-torus ((III)IIII) - S4xS2
87. 214-ditorus (((II)I)IIII) - S4xT2
88. 2221-triger ((II)(II)(II)I) - S3xC3
89. 223-tiger ((II)(II)III) - S4xC2
90. 25-torus ((II)IIIII) - S5xS1

EDIT 7/5 : Minor corrections and additions
Last edited by ICN5D on Sun Jul 06, 2014 3:32 am, edited 3 times in total.
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Re: The Tiger Explained

Postby Marek14 » Sat Jul 05, 2014 4:20 am

I wonder -- when you change the order of factors on right, will you get a toratope that is topologically equivalent?
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Re: The Tiger Explained

Postby ICN5D » Sat Jul 05, 2014 5:28 am

As in,

S2xS1xC2 = S2x[T2*S1] = S2x[(S1xS1)*S1] = S2x[S1*S1]xS1 = S2xS1x[S1*S1] ?

They're all equivalent. It depends on inflation sequence. This system actually follows my notation for toratopes, strangely enough. But, I wouldn't have known about the commutative property for a while. It happens with higher order Clifford tori, as in the tori or other inflations of the original duocylinder margin. So, for a duoring torus: Circle along circle times circle, [(S1xS1)*S1] equals circle times circle along circle, [S1*S1]xS1 the 3-manifold edge of a cyltorinder, aka duocylinder torus.

Edit:

But, if you mean S1xS2xS3 , then no it will not equal S1xS3xS2 or S2xS1xS3 or S3xS1xS2 or S3xS2xS1. An n-sphere bundle can only commute in and out of clifford manifold products, the edges of inflation.
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Re: The Tiger Explained

Postby Marek14 » Sat Jul 05, 2014 6:26 am

ICN5D wrote:As in,

S2xS1xC2 = S2x[T2*S1] = S2x[(S1xS1)*S1] = S2x[S1*S1]xS1 = S2xS1x[S1*S1] ?

They're all equivalent. It depends on inflation sequence. This system actually follows my notation for toratopes, strangely enough. But, I wouldn't have known about the commutative property for a while. It happens with higher order Clifford tori, as in the tori or other inflations of the original duocylinder margin. So, for a duoring torus: Circle along circle times circle, [(S1xS1)*S1] equals circle times circle along circle, [S1*S1]xS1 the 3-manifold edge of a cyltorinder, aka duocylinder torus.

Edit:

But, if you mean S1xS2xS3 , then no it will not equal S1xS3xS2 or S2xS1xS3 or S3xS1xS2 or S3xS2xS1. An n-sphere bundle can only commute in and out of clifford manifold products, the edges of inflation.


I didn't mean equal, I meant topologically equivalent. For example, torisphere and spheritorus both have S1xS2 topology. There's a way to turn a torus "inside out" in 3D -- if you try this on torisphere or spheritorus, they should transform in the other toratope.
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Re: The Tiger Explained

Postby ICN5D » Sat Jul 05, 2014 2:00 pm

It seems that the order of S1xS2 shows the toratopic dual relationships, when reversed. Is that what you mean? What exactly does 'topological equivalent' mean?
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Re: The Tiger Explained

Postby Marek14 » Sat Jul 05, 2014 2:51 pm

ICN5D wrote:It seems that the order of S1xS2 shows the toratopic dual relationships, when reversed. Is that what you mean? What exactly does 'topological equivalent' mean?


Topological equivalent means, simply said, that if you were confined on a surface of a toratope, you wouldn't know which one it is.

Both torisphere and spheritorus have S1xS2 as their surface -- you'd have nonvanishing spheres in two dimensions and nonvanishing circles in the third dimension.
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Re: The Tiger Explained

Postby ICN5D » Sat Jul 05, 2014 3:09 pm

Okay, I see how that works. Other instances also seem to relate to a commuting of Sn, Tn in and out of a Clifford manifold.
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Re: The Tiger Explained

Postby Marek14 » Sat Jul 05, 2014 3:36 pm

ICN5D wrote:Okay, I see how that works. Other instances also seem to relate to a commuting of Sn, Tn in and out of a Clifford manifold.


Well, actually I think that all toratopes might topologically be some combinations of S spaces. For example, duocylinder margin has dimension S1xS1 (since it's product of two S1 spaces), the same as torus -- so that would mean that ditorus and tiger have the same topology S1xS1xS1.
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Re: The Tiger Explained

Postby ICN5D » Sat Jul 05, 2014 3:44 pm

I would agree with that. That's what I like about this system, as it shows up quite well. I'm inclined to enumerate 8D this evening.
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Re: The Tiger Explained

Postby Marek14 » Sat Jul 05, 2014 4:32 pm

ICN5D wrote:I would agree with that. That's what I like about this system, as it shows up quite well. I'm inclined to enumerate 8D this evening.


Good luck with that :) In 8 and more D, it starts to get tiresome.
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Re: The Tiger Explained

Postby ICN5D » Sun Jul 06, 2014 3:13 am

Well, actually, now that I've thought about it, wouldn't [S1*S1] be different from S1xS1 ? Both are products of two circles, but they differ in orientation. The [S1*S1] is the duocylinder margin C2 from [B2*B2] ( Bn = n-ball); and S1xS1 is the torus T2. But, I think I remember seeing somewhere that a ditorus T3 and tiger S1xC2 are topological equivalents. Both are a product of three circles, but they differ in orientation. Where if on the surface of a ditorus, wouldn't we see the effects of the medium diameter, which is not present on a tiger? If travelling around the diameters of the 3-manifold, we could trace out three different distances that repeat, in either X, Y, or Z. If measuring the diameters on the 3-manifold of a tiger, we would record two equal large and one small repeating lengths, in X, Y, or Z.
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Re: The Tiger Explained

Postby Marek14 » Sun Jul 06, 2014 4:56 am

ICN5D wrote:Well, actually, now that I've thought about it, wouldn't [S1*S1] be different from S1xS1 ? Both are products of two circles, but they differ in orientation. The [S1*S1] is the duocylinder margin C2 from [B2*B2] ( Bn = n-ball); and S1xS1 is the torus T2. But, I think I remember seeing somewhere that a ditorus T3 and tiger S1xC2 are topological equivalents. Both are a product of three circles, but they differ in orientation. Where if on the surface of a ditorus, wouldn't we see the effects of the medium diameter, which is not present on a tiger? If travelling around the diameters of the 3-manifold, we could trace out three different distances that repeat, in either X, Y, or Z. If measuring the diameters on the 3-manifold of a tiger, we would record two equal large and one small repeating lengths, in X, Y, or Z.


The thing is that distances play no part in topology.
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Re: The Tiger Explained

Postby ICN5D » Sun Jul 06, 2014 5:47 am

I couldn't resist using this bundle over n-sphere terminology to analyze something more complex, and I found some really neat patterns in 9D. Check it out, and I'm sure there's more than this:


The three distinct toroids from (((II)I)(II)) bundle over (((II)I)(II)), the 8-manifold XYZWVUTS:



Bn - n-Ball, solid n-sphere
Rn - n-Ring , solid n-torus structure , Rn = B2 x T(n-1) , T1 = S1
Sn - n-Sphere, D-1 surface of hollow n-ball
Tn - n-Torus, D-1 surface of hollow n-ring
Cn - n-Clifford manifold, edge of (B2^n) prism as [S1^n] margin
[(Sn/Tn x Cn x Sn)*Sn] - n-Clifford manifold, edge of (B(n+1)/R(n+1) x Cn x Sn)*B(n+1) open toratope prism




• (((!I)I)(II)) --> (((II)I)(II)) = ((((((II)I)(II))I)I)(II)) = ((((((maj1)sec)(maj2)tert)quat)quint)(maj3)min)
-Tigroid bundle over tigroid:
S1x[T2*S1]xS1x[T2*S1] = S1x[(S1xS1)*S1]xS1x[(S1xS1)*S1]

-Clifford manifold combos:
S1xC2xT2xC2xS1 = S1x[S1*S1]xS1xS1x[S1*S1]xS1
T5xC3 = S1xS1xS1xS1xS1x[S1*S1*S1]

-Pure tigroid expression:
S1x[(T3xC2xS1)*S1] = S1x[(S1xS1xS1x[S1*S1]xS1)*S1]

[(T3xC2xS1)*S1] is the 7D margin of a (R4xC2xS1)*B2 prism , (((((II)I)(II))I)I)(II)




• (((II)!)(II)) --> (((II)I)(II)) = (((((II)I)(II))(II))(II)) = (((((maj1)sec)(maj2)tert)(maj3)quat)(maj4)min)
-Tigroid bundle over tigroid:
S1x[T2*S1]xS1x[T2*S1] = S1x[(S1xS1)*S1]xS1x[(S1xS1)*S1]

-Clifford manifold combos:
S1xC2xC2xC2xS1 = S1x[S1*S1]x[S1*S1]x[S1*S1]xS1
T3xC4xS1 = S1xS1xS1x[S1*S1*S1*S1]xS1
T5xC3 = S1xS1xS1xS1xS1x[S1*S1*S1]

-Pure tigroid expression:
S1x[(S1xC2xC2xS1)*S1] = S1x[(S1x[S1*S1]x[S1*S1]xS1)*S1]

[(S1xC2xC2xS1)*S1] is the 7D margin of a (B2xC2xC2xS1)*B2 prism , ((((II)I)(II))(II))(II)






• (((II)I)(!I)) --> (((II)I)(II)) = (((((II)I)(II))I)((II)I)) = (((((maj1)sec1)(maj2)tert)quat)((maj3)sec2)min)
-Tigroid bundle over tigroid:
S1x[T2*S1]xS1x[T2*S1] = S1x[(S1xS1)*S1]xS1x[(S1xS1)*S1]

-Clifford manifold combos:
S1xC2xT2xC2xS1 = S1x[S1*S1]xS1xS1x[S1*S1]xS1
S1xC2xT2xC3 = S1x[S1*S1]xS1xS1x[S1*S1*S1]

-Pure tigroid expression:
S1x[(T2xC2xS1)*T2] = S1x[(S1xS1x[S1*S1]xS1)*(S1xS1)]


[(T2xC2xS1)*T2] is the 7D margin of a (R3xC2xS1)*R3 prism , ((((II)I)(II))I)((II)I)
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Re: The Tiger Explained

Postby Marek14 » Sun Jul 06, 2014 6:33 am

Looks interesting :)
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Re: The Tiger Explained

Postby ICN5D » Mon Jul 07, 2014 2:45 am

Something I haven't tried until now was nailing down a good conversion procedure, for the implicit functions from toratope notation. I used the ones you posted and changed them a little bit, for other shapes. But, I didn't really try to convert it from scratch. This is detailed in the Wiki, but I didn't understand how until now. I played around with many different orders, and found that by leaving the dimensions unsquared until the last step helped with setting up the square roots.



(((((II)I)(II))I)((II)I)) = 0

(((((xy)z)(wv))u)((ts)r)) = 0

((((xy)z)(wv))u)((ts)r) = 0

(( ((xy)z) (wv) ) u) ((ts)r) = 0

(( ((x+y)+z) + (w+v) ) +u) + ((t+s)+r) = 0

(( ((x+y -R1)+z -R2) + (w+v -R3) -R4) +u -R5) + ((t+s -R6)+r -R7) -R8 = 0

(( ((x+y -R1)^2 +z -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0

(( ((sqrt(x+y) -R1)^2 +z -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0

(( (sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0

(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + ((sqrt(t+s) -R6)^2 +r -R7)^2 -R8^2 = 0

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + (sqrt((sqrt(t+s) -R6)^2 +r) -R7)^2 -R8^2 = 0

(sqrt(((sqrt((sqrt(x^2+y^2) -R1)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R3)^2 -R4)^2 +u^2) -R5)^2 + (sqrt((sqrt(t^2+s^2) -R6)^2 +r^2) -R7)^2 -R8^2 = 0




(((((II)I)(II))I)((II)I)) = 0


1. write dimension letter

(((((xy)z)(wv))u)((ts)r)) = 0


2. remove outermost ()

((((xy)z)(wv))u)((ts)r) = 0


3. separate distinct toratope parameters

(( ((xy)z) (wv) ) u) ((ts)r) = 0


4. add '+' between dimensions and parameters

(( ((x+y)+z) + (w+v) ) +u) + ((t+s)+r) = 0


5. add '-Rn' to left of close parentheses AND the minor R at the end : [... ) ...] -> [... -Rn) ...]

(( ((x+y -R1)+z -R2) + (w+v -R3) -R4) +u -R5) + ((t+s -R6)+r -R7) -R8 = 0


6. add '^2' to right of close parentheses AND minor diameter : [... Rn) ...] --> [... Rn)^2 ...] , Rminor --> Rminor^2

(( ((x+y -R1)^2 +z -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0


7. add sqrt(...) around X+Y : [... (x+y-R1) ...] --> [... (sqrt(x+y)-R1) ...]

(( ((sqrt(x+y) -R1)^2 +z -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0


8. add (sqrt...) around closed set with Z : [...((...)^2 +z...] --> [...(sqrt((...)^2 +z)...]

(( (sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (w+v -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0


9. add sqrt(...) around W+V : [...(w+v-R3)...] --> [...(sqrt(w+v)-R3)...]

(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0


10. add (sqrt...) around entire closed set with U : [ (((...)^2 +u ...] --> [ (sqrt(((...)^2 +u) ...]

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + ((t+s -R6)^2 +r -R7)^2 -R8^2 = 0


11. add sqrt(...) around T+S : [...(t+s-R6)...] --> [...(sqrt(t+s)-R6)...]

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + ((sqrt(t+s) -R6)^2 +r -R7)^2 -R8^2 = 0


12. add (sqrt...) around closed set with R : [...((...)^2 +r...] --> [...(sqrt((...)^2 +r)...]

(sqrt(((sqrt((sqrt(x+y) -R1)^2 +z) -R2)^2 + (sqrt(w+v) -R3)^2 -R4)^2 +u) -R5)^2 + (sqrt((sqrt(t+s) -R6)^2 +r) -R7)^2 -R8^2 = 0


13. add '^2' to all dimensions

(sqrt(((sqrt((sqrt(x^2+y^2) -R1)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R3)^2 -R4)^2 +u^2) -R5)^2 + (sqrt((sqrt(t^2+s^2) -R6)^2 +r^2) -R7)^2 -R8^2 = 0





(sqrt(((sqrt((sqrt(x^2+y^2) -R1)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R3)^2 -R4)^2 +u^2) -R5)^2 + (sqrt((sqrt(t^2+s^2) -R6)^2 +r^2) -R7)^2 -R8^2 = 0


And, voila! There's an implicit function that can be used to explore the shape in great detail!
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Re: The Tiger Explained

Postby Marek14 » Mon Jul 07, 2014 5:48 am

You could have mentioned that before :) I think I've been using something similar from the start.
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Re: The Tiger Explained

Postby ICN5D » Mon Jul 07, 2014 7:20 am

Well, you gave me so many equations from the start! It was easy to build new ones off of them, I never thought about it. But, this little exercise was really good for my own benefit. I'm going to practice it a bunch, for sure. I read in the wiki that you detailed the conversion a while back, but I never really attempted it before.

By the way, I put together the function for ((((II)I)((II)I))I), and wow! There's some really amazing stuff going on. Just explored it a little tonight:


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



This one is a new type of function, with adjustable diameters. I'm going to use this from now on, to test a new equation, and get Rn values easily.
• ((((I))((I)))I) - 16 tori in 4x4x1 maj array - Diameter Adjustment Equation
((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

Maj1,2 - A
Sec1,2 - B
Tertiary - C
Minor - D




Axial translate of trace cut
• ((((I))((I)))I) - 16 tori in 4x4x1 maj array
((sqrt((sqrt(x^2+a^2) -5)^2 +b^2) -2.5)^2 + (sqrt((sqrt(y^2+c^2) -5)^2 +d^2) -2.5)^2 -2)^2 +z^2 -1.25^2 = 0




Neat Exploration Functions

• ((((Xy)z)((Yc)x))Z) - ((((I))((I)))I)
((sqrt((sqrt((x*sin(d))^2+(y*cos(a))^2) -5)^2 +(z*cos(b))^2) -2.5)^2 + (sqrt((sqrt((y*sin(a))^2+c^2) -5)^2 +(x*cos(d))^2) -2.5)^2 -2)^2 +(z*sin(b))^2 -1.25^2 = 0

--- B=1.57/C,D=0 : Adjust A for rotate past 4x diagonal (((II)I)I) in 2x1x4x1 vert rectangle of 8 ditoruses ((((I)I)((I)))I)



• ((((Ac))((C)a))I) - ((((I))((I)))I)
((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(d) - c*sin(d))^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2+0^2) -5)^2 +(x*cos(b) - a*sin(b))^2) -2.5)^2 -2)^2 +z^2 -1.25^2 = 0

--- [A,B]=[0,1.57] : Animate C=-12.5,+12.5 , Step Adjust D in 6 Increments between 0,1.57, after each cycle of C
--- [A,B]=[0,1.57] / C=2.5 : Animate D , and set C=7.5 anim D for tritorus empty cut morphing

^^^ This one I HAVE to animate before anything else. It'll require 5-6 of them to show the amazing change in action. It's totally amazing.
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