The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby Marek14 » Mon Jun 16, 2014 7:25 am

ICN5D wrote:Yes it does :) It's really close to the oblique cut (((O)I)((O)I)) , between (((I)I)(()I)) and ((()I)((I)I)). The two tiger cages oblique. I've got an animation coming of that one, too! I rotated to 45 degrees between the square of four tigers (((I)I)((I)I)) , and translated out. Moving past the square at a 45 degree angle is super cool to see a 1x2x1 oblique evolution of the four tigers, cutting the square like a diamond. Again, that new function is absolutely amazing. One can actually pilot a 3D hyperplane around this way, without being tethered to the origin. Like steering an extra-dimensional spaceship, it's way cool going out to survey geometric structures in their home dimension :) Just wait, I'm cooking up some insane animation ideas .....


That sounds great :) I'm glad I could help :)
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 17, 2014 5:48 am

Check 'em out:


For all: set B,D = 0 ~ 1.57



Surveying (((II)I)((II)I)) :

Lets you do oblique translate of amazing quad tiger cage of (((II)I)((II)I))

• (((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, the quad tiger cage
-- XYZ = -5/+5
-- A = -4.5~4.5


Lets you explore the 4 Tigers in the flat square arrangement (((I)I)((I)I))

• (((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





Surveying ((((II)I)(II))I) :

Lets you explore the vertical square of four ditoruses ((((I)I)(I))I)

• ((((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



Alternate Rotate + Translate within vert square, shows very interesting things when translating out of empty cuts to a shape, then do rotations

• ((((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
-- Extremely amazing exploration of the vert square of ditoruses. I was fascinated by this diagonal rotation, and had to check it out more closely

For something cool:
-- A= -1.29 , C= 0.9 : Adjust B,D angles through all four combinations, very cool stuff!







I can't help but notice using a systematic way for building equations, based on toratope notation, is evolving the notation. In order to keep track of equation forms and set up their function ahead of time, I have developed on it a little. Ultimately, we use it to derive the actual equations, and this adaptation can represent them:


For (((II)I)((II)I)),

(((I)I)((I))) - static axial cut

(((OI))((OI))) - static 45 deg oblique cut, " O " is between both axial positions, as a means of representing derivable cool ones

(((Ii)I)((Ii)i) - dynamic axial translate evolution

(((X)I)((Ix))) - dynamic single rotation, start at uppercase, end at lowercase, for 90 degree rotation using 0~1.57

(((X)Y)((Ix)y)) - dynamic double rotation

(((Xz)Y)((Zx)y)) - dynamic triple rotation

(((A)I)((Ca)c)) - dynamic rotate + translate , Axis 1: translate A , rotate B // Axis 2: translate C, rotate D
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Re: The Tiger Explained

Postby Marek14 » Tue Jun 17, 2014 6:58 am

Hm, were there any images in that last post? Can't see them...
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 17, 2014 2:04 pm

no, not yet. exhausted after my new 80 mile daily commute
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Re: The Tiger Explained

Postby ICN5D » Thu Jun 19, 2014 3:27 am

Okay, here it is! The amazing, incredible transformations of that unique empty to empty rotation in ((((II)I)(II))I). I am very curious about this region, and had to take a closer look.


Remember this rotation montage? It's scanning past two ditoruses out of four, stacked in a maj/med vertical square arrangement : ((((I)I)(I))I) . Which makes this rotation ((((X)I)(x))I) . Our 3D plane has slid between the four, and we are rotating from the horizontal empty to the vertical empty. In doing so, we scan past the diagonally opposed:

Image




In the animation below, I used the equation form ((((A)C)(ac))I) ,which converts to

((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

Set B,D = 0 ~ 1.57 , for 90 degree rotations

Setting A = -1.29 , C = 0.9 , and going through all four combinations of 90 degree rotations, in the XY hyperplane: [0,0] - [0,1.57] - [1.57,0] - [1.57,1.57], but not in this particular order. It's more like a cycle of [0,0] - [1.57,0] - [1.57,1.57] - [0,1.57] - [0,0] .




So, basically, what this function does, is let you completely explore the square array. I believe there are 3-4 combinations of this function. Adjusting A and C will translate out to the shapes, B and D will rotate in those higher dimensions. The special coordinates of A and C place us in a unique position, slicing the two ditoruses. We see a very complex topology change with each 90 deg rotation, leading back to the beginning. Which is awesome.



Image


Is that not the wildest morphing you've ever seen? Wow! There's a lot more to see in this array, too, in showing all the different scans. More to come on this one ....
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Re: The Tiger Explained

Postby ICN5D » Sat Jun 21, 2014 5:02 am

Just got done with making all Ditorus animation scenes, Villarceau sections, and lemniscates. Will post tomorrow! Cool stuff!
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Re: The Tiger Explained

Postby Marek14 » Sat Jun 21, 2014 6:05 am

ICN5D wrote:Just got done with making all Ditorus animation scenes, Villarceau sections, and lemniscates. Will post tomorrow! Cool stuff!


And I'm making some interesting progress with chain exploration :)
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Re: The Tiger Explained

Postby Marek14 » Sat Jun 21, 2014 12:34 pm

I got an interesting line of thought, but can't find the proper equation.

It goes like this:
Tiger can be thought as having four "legs", that intersect plane as four circles, and evolve into vertical stack of toruses in two different ways. If we put them in vertices of rectangle, one evolution joins circles "horizontally" and the other one "vertically".

What if we have more than four circles?

If we put six circles in vertices of hexagon (not necessarily regular, it could have two alternating edge lengths), there should exist an analogical shape to tiger where one evolution joins circles 1-2, 3-4 and 5-6 and the other circles 2-3, 4-5 and 6-1. And analogically for other even polygons.

If 4-legged beast is tiger, then 6-legged would be the MANTIS, 8-legged the SPIDER and 10-legged the CRAB.

But so far I can't find the proper equation. Equation for the planar cut, i.e. the circles in vertices of polygon, would be the start. In tiger case, the equation for four circles is (sqrt(x^2) - a)^2 + (sqrt(y^2) - b)^2 = c. In polar coordinates this would be (sqrt((r*cos(f))^2) - a)^2 + (sqrt((r*sin(f))^2) - b)^2 = c. Now, I realized that if we put this as square and reduce circles to points, (sqrt((r*cos(f))^2) - a)^2 + (sqrt((r*sin(f))^2) - a)^2 = 0, it will be equivalent to (r*cos(f))^2 = a^2 and (r*sin(f))^2 = a^2, so cos(f)^2 = sin(f)^2, which is equivalent to cos(f)^2 - sin(f)^2 = 0, and thus cos(2f) = 0. So maybe the mantis would have cos(3f) = cos(f)^3 - 3*sin(f)^2*cos(f) present somewhere...
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 22, 2014 10:41 pm

Six legged beasts? Hmm, sounds interesting! I thought it could exist, too. Could there be any relation to what is named a triple torus, with three handles? Maybe these are tiger related versions, which have undergone a non-intersecting rotation around the minor diameter hyperplane?


Triple Torus

Image


These could be genus three tigers, or genus four!
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 23, 2014 1:26 am

All right everyone, presenting,



4D Ditorus : (((II)I)I)


(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0



A ditorus can be made three ways through inflation:

• circle --> circle --> circle : three iterations of inflating the 1D edge of a circle with another
• circle --> torus : inflating the 2D skin of a torus with a circle
• torus --> circle : inflating the 1D edge of a circle with a 3D torus


The diameter hierarchy of a ditorus goes by : (((maj)med)min)




Axial Translations



(((I)I)I) - 2 displaced torii along a line

(sqrt((sqrt(x^2 + a^2) - 2.5)^2 + y^2) - 1)^2 + z^2 -0.5^2 = 0

Image




(((II))I) - concentric major pair of 2 torii

(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + a^2) - 1)^2 + z^2 -0.5^2 = 0

Image




(((II)I)) - concentric minor pair of 2 torii

(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + z^2) - 1)^2 + a^2 -0.5^2 = 0

Image







Axial Rotations



(((IY)y)I) : displaced to concentric major pair rotation

(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + z^2 - 0.5^2 = 0

Image




(((II)Z)z) : concentric major pair to minor pair rotation

(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + (z*sin(a))^2) - 1)^2 + (z*cos(a))^2 - 0.5^2 = 0

Image




(((IY)I)y) - displaced to concentric minor pair rotation

(sqrt((sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2) - 1)^2 + (y*cos(a))^2 - 0.5^2 = 0

Image





Lemniscates of a Ditorus

There are four distinct lemniscates of a ditorus, where three of which come from the displaced torus translation (((Ii)I)I), and only one from the concentric major pair translate (((II)i)I) .


Ditorus Lemniscate A

Made from the axial translation of (((Ii)I)I), as two cassini deforming displaced torii merge/split

Image



Ditorus Lemniscate B

The second instance of a lemniscate from the displaced torus evolution (((Ii)I)I)

Image



Ditorus Lemniscate C

The third lemniscate from the (((Ii)I)I) translation

Image



Ditorus Lemniscate D

From the axial translation of (((II)i)I), as two cassini deforming major concentric torii merge/split. A lemniscate does not exist for a minor pairing, being analogical to the concentric circle evolution of a torus : ((II)i)

Image






Villarceau Sections of a Ditorus

There are five distinct bitangent sections of a ditorus, where two exist for the rotations (((IY)y)I) and (((IY)I)y) each.



Villarceau section 1A

The first instance of a bitangent cut from rotation (((IY)y)I)

Image



Villarceau section 1B

The second bitangent cut from rotation (((IY)y)I)

Image



Villarceau section 2A

First bitangent cut from rotation (((IY)I)y)

Image



Villarceau section 2B

Second bitangent cut from rotation (((IY)I)y), trimmed open to reveal the contact points inside. Notice the footprint is very similar to the villarceau section of a torus, as is the case with all of them, taking various forms.

Image



Villarceau section 3

A very special kind, which is bitangent to not just two points, but two whole 1D edges of a circle ! How about that? A multidimensional Villarceau section, tangent to a infinite number of points along a curving line, the first case of which shows up in a ditorus. From rotation (((II)Z)z), going from a major to minor pair, as the walls creep up and close over, similar to the above 2A and 2B. Also noteworthy, is the nature of this being two displaced V-sections of a torus.

Image




EDIT 6/23: Added the fourth lemniscate of the hole closing up during translation (((Ii)I)I), as pointed out by Marek.
Last edited by ICN5D on Tue Jun 24, 2014 2:03 am, edited 2 times in total.
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Re: The Tiger Explained

Postby Marek14 » Mon Jun 23, 2014 4:54 am

There should be one extra lemniscate, though, in two torus cut, when the hole finally closes up completely.
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 23, 2014 1:53 pm

I was wondering if that one counted. Guess so! I'll add it thia evening.
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Re: The Tiger Explained

Postby Marek14 » Mon Jun 23, 2014 3:42 pm

Well, why wouldn't it count? It's a topology change :)
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 24, 2014 2:10 am

Updated, thanks! I'll look out for those in the future. Are there any other noteworthy things I should showcase for future toratopes? This last one is the most complete out of all others that have come before. In fact, this thread from page 4 to 21 is a complete record of the evolution in my understanding! Beginning with the first post about what tiger is, all the way through learning toratope notation, and now CalcPlot renders and animations, which have gotten rather sophisticated now.


Now I'm wondering, which one to do next?
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Re: The Tiger Explained

Postby Marek14 » Tue Jun 24, 2014 4:46 am

ICN5D wrote:Updated, thanks! I'll look out for those in the future. Are there any other noteworthy things I should showcase for future toratopes? This last one is the most complete out of all others that have come before. In fact, this thread from page 4 to 21 is a complete record of the evolution in my understanding! Beginning with the first post about what tiger is, all the way through learning toratope notation, and now CalcPlot renders and animations, which have gotten rather sophisticated now.


Now I'm wondering, which one to do next?


Well, now we could go to 5D, but to do so, it will be useful to get to the arrays. You could try to make a snapshot in two dimensions of parameters, reduce the individual pictures in size, and combine them into tapestry. And if you try that, then the first toratope should be the pentasphere, as a zero element.
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Re: The Tiger Explained

Postby Marek14 » Tue Jun 24, 2014 9:43 am

OK, here's the first (quite imperfect) spider:

(sqrt((x^2 - y^2)^2/(x^2 + y^2) + z^2) - 2)^2 + (sqrt((x*y)^2/(x^2 + y^2) + w^2) - 2)^2 = 1/2

I created it by expressing the equation for four circles in polar coordinates and then replaced fi with 2*fi, transformed back into cartesian coordinates and added z and w back. The shape isn't yet perfect, but the topology can be seen.

As for 5D toratopes, following images might be interesting:

The low-res 2D arrays. Note that there are two types of arrays: if you remove two I's from the same place, you'll get an array with circular symmetry which can be shown as concentric circles of identical images. If you remove two I's from different places, you'll get an array that's best shown as rectangle.

Lemniscates/singularities: Generally, a curve in the 2D array will be formed by lemniscates. One possibility would be to map that curve and show an animation that passes through all lemniscates in the given representation. This only makes sense for rectangular arrays, though -- in circular arrays the curve is circle and the lemniscate would not change at all as you'd trace it.

Coordinate rotations and double rotations - but you should make sure of not needlessly remaking rotations that were already present in lower D.

So, let's look once again at the 2D arrays lemniscate morphs:

Pentasphere (IIIII)
One circular array, (IIIii).

41-torus ((IIII)I)
One circular array, ((IIii)I).
One rectangular array, ((IIIi)i). Not really suitable for lemniscate morphs since it involves shapes like sphere plus one point in center.

311--ditorus ((III)I)I)
One circular array, ((Iii)I)I).
Three rectangular arrays, ((III)i)i), ((IIi)I)i) and ((IIi)i)I). The first has bad morphs (involving various coinciding spheres). Ditoruses in general have three singular points when evolved in major dimension and one when evolved in medium dimension, so the other two arrays might have interesting lemniscates.

Tritorus ((II)I)I)I)
One circular array, ((ii)I)I)I).
Six rectangular arrays, ((II)I)i)i), ((II)i)I)i), ((Ii)I)I)i), ((II)i)i)I), ((Ii)I)i)I) and ((Ii)i)I)I). For tritorus, major evolution has seven singular points, secondary has three and tertiary has one.

Tiger torus (((II)(II))I)
One circular array, ((II)(ii))I).
Two rectangular arrays, (((II)(Ii))i) and (((Ii)(Ii))I). Tiger torus has three singular points in major evolution.

221-ditorus (((II)II)I)
Two circular arrays, (((II)ii)I) and (((ii)II)I).
Three rectangular arrays, (((II)Ii)i), (((Ii)II)i) and (((Ii)Ii)I).

320-tiger ((III)(II))
Two circular arrays, ((III)(ii)) and ((Iii)(II)).
One rectangular array, ((IIi)(Ii)). Tiger has one singular point in major evolution.

Torus tiger (((II)I)(II))
Two circular arrays, (((II)I)(ii)) and (((ii)I)(II)).
Three rectangular arrays, (((II)i)(Ii)), (((Ii)I)(Ii)) and (((Ii)i)(II)). Torus tiger has three singular point in major[torus] evolution and one in medium and major[circle] evolution. In general, number of singular points is 2^(n-1) - 1 where n is the "nesting level" of the given dimension in representation. One extra singular point can be taken as the border of toratope where it vanishes.

32-torus ((III)II)
Two circular arrays, ((III)ii) and ((Iii)II).
One rectangular array, ((IIi)Ii).

212-ditorus (((II)I)II)
Two circular arrays, (((II)I)ii) and (((ii)I)II).
Three rectangular arrays, (((II)i)Ii), (((Ii)I)Ii) and (((Ii)i)II).

221-tiger ((II)(II)I)
One circular array, ((II)(ii)I).
Two rectangular arrays, ((II)(Ii)i) and ((Ii)(Ii)I).

23-torus ((II)III)
Two circular arrays, ((II)Iii) and ((ii)III).
One rectangular array, ((Ii)IIi).
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Re: The Tiger Explained

Postby Marek14 » Wed Jun 25, 2014 7:10 pm

I compiled a full list of 261 8D toratopes with my names:
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Re: The Tiger Explained

Postby Marek14 » Thu Jun 26, 2014 8:08 am

I made some corrections to 8D list and compiled the full list of 766 9D toratopes.

Bonus homework: compile the cut list for the beast with number 666, the torus tiger triger ((((II)I)(II))(II)(II)). Its trace is 4x2x2x2 array of spheritoruses.
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Re: The Tiger Explained

Postby ICN5D » Thu Jun 26, 2014 1:54 pm

Awesome, look at that huge list. It's fun to scan over it and do mental cuts in your head! It has that " oooh shiny object" appeal, you know?
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Re: The Tiger Explained

Postby Marek14 » Thu Jun 26, 2014 2:40 pm

Yeah, that took about hour or two to write :)
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Re: The Tiger Explained

Postby wendy » Thu Jun 26, 2014 11:13 pm

A tiger is a torus swirlprism.

Swirlprism is in essence a transform from a 3d space into 4d cliffords. So if you make a icosahedron edgeframe and make it a swirl prism, you would get a genus 11 tiger.

You can make one by hooking up the dodecahedra of a hollow twelftycoron into twelve loops of ten.

This is beyond the scope of rototope notation, though.
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Re: The Tiger Explained

Postby ICN5D » Sat Jun 28, 2014 5:18 am

I have to go to bed soon, so here is most the unnamed list of the cuts of ((((II)I)(II))(II)(II)) . I still have yet to do the last bit in 4D, just realized after posting :oops: . That took a little bit of time in itself, just to enumerate them and delete duplicates! I've been trying to break them down by diameter hierarchy lately, using the inflation process as a reference. So, apparently, ((((II)I)(II))(II)(II)) has one major, two equal secondaries, one tertiary, two equal quaternaries, and a minor diameter. Keeping all of the empty cuts makes the combinatorics more clear. Next step is a description of the cuts ....



9D
((((II)I)(II))(II)(II)) - sphere --> trioring --> duoring --> ring , ((((maj)sec1)(sec2)tert)(quat1)(quat2)min)
---------------------------------------------------------------------------------------------------------------
8D
((((I)I)(II))(II)(II))
((((II))(II))(II)(II))
((((II)I)(I))(II)(II))
((((II)I)(II))(I)(II))
-------------------------------------------------------------------------------
7D
(((()I)(II))(II)(II))
((((I))(II))(II)(II))
((((I)I)(I))(II)(II))
((((I)I)(II))(I)(II))
((((II))(I))(II)(II))
((((II))(II))(I)(II))
((((II))(I))(II)(II))
((((II)I)())(II)(II))
((((II)I)(I))(I)(II))
((((II)I)(II))()(II))
((((II)I)(II))(I)(I))
--------------------------------------------------------------------------------
6D
(((())(II))(II)(II))
(((()I)(I))(II)(II))
(((()I)(II))(I)(II))
--------------------
((((I))(I))(II)(II))
((((I))(II))(I)(II))
((((I)I)())(II)(II))
((((I)I)(I))(I)(II))
((((I)I)(II))()(II))
((((I)I)(II))(I)(I))
--------------------
((((II))())(II)(II))
((((II))(I))(I)(II))
((((II))(II))()(II))
((((II))(II))(I)(I))
((((II)I)())(I)(II))
((((II)I)(I))()(II))
((((II)I)(I))(I)(I))
((((II)I)(II))()(I))
-----------------------------------------------------------------------------------------
5D
(((())(I))(II)(II))
(((())(II))(I)(II))
(((()I)())(II)(II))
(((()I)(I))(I)(II))
(((()I)(II))()(II))
(((()I)(II))(I)(I))
-------------------
((((I))())(II)(II))
((((I))(I))(I)(II))
((((I))(I))(II)(I))
((((I))(II))()(II))
((((I))(II))(I)(I))
((((I)I)())(I)(II))
((((I)I)(I))()(II))
((((I)I)(I))(I)(I))
((((I)I)(II))()(I))
-------------------
((((II))())(I)(II))
((((II))(I))()(II))
((((II))(I))(I)(I))
((((II))(II))()(I))
((((II)I)())()(II))
((((II)I)())(I)(I))
((((II)I)(I))()(I))
((((II)I)(II))()())
----------------------------------------------------------------------------------------------
4D
(((())())(II)(II))
(((())(I))(I)(II))
(((())(II))()(II))
(((())(II))(I)(I))
(((())(I))(II)(I))
------------------
(((()I)())(I)(II))
(((()I)(I))()(II))
(((()I)(I))(I)(I))
(((()I)(II))()(I))
------------------
((((I))())(I)(II))
((((I))(I))(I)(I)) * 4x2x2x2 array of spheritoruses
((((I))())(II)(I))
((((I))(I))(II)())
((((I))(II))()(I))
((((I)I)())()(II))
((((I)I)())(I)(I))
((((I))(I))()(II))
((((I)I)(I))()(I))
((((I)I)(II))()())


And then the rest, beginning with ((((II) ......
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Re: The Tiger Explained

Postby Marek14 » Sat Jun 28, 2014 5:42 am

You're doing fine :)
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Re: The Tiger Explained

Postby Secret » Sat Jun 28, 2014 7:30 am

ICN5D
Love those gifs


Now I can double check my visualisation progress and see if there are errors in my drawings

I am miles behind this thread now due to uni and stuff, it might took a while to catch up and start makings cuts with you guys
Meanwhile I have this question
I am still at lost on how the duoring is cut in order to give you the "Tiger Cage"

Is it somehting like what is shwon here
P.S. The other drawings is trying to visualise the shape based on the gif and previous understanding of them, and more like scribbles can be ignored
Capturooe3.PNG
(854.5 KiB) Not downloaded yet
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Re: The Tiger Explained

Postby Marek14 » Sat Jun 28, 2014 8:16 am

Secret: The standard duoring is placed in coordinate planes, so one ring is in xy plane and the other is in zw plane. To get the cage, you have to rotate it 45 degrees in one of the "cross-planes", i.e. xz, xw, yz or yw. Alternately, you slice normal duoring with hyperplane whose normal is halfway between one of these pairs of directons.

Normal duoring is a set of points that satisfy two equations:

x^2 + y^2 = r^2
z^2 + w^2 = r^2

Both radii are the same here -- otherwise we'd have to rotate it by some other angle to get the cage.

Now, rotating by 45 degrees in yw plane is equivalent to replacing y^2 with (y + w)^2/2 and w^2 with (y - w)^2/2

x^2 + (y + w)^2/2 = r^2
z^2 + (y - w)^2/2 = r^2

In xyw hyperplane or yzw hyperplane we'll get two circles, just not in coordinate planes. But in xyz and xyw hyperplanes, it will simplify differently:

w = 0:
x^2 + y^2/2 = r^2
z^2 + y^2/2 = r^2

Since both equations must be satisfied at once, this means that either x = z or x = -z. Each option traces an ellipse in 3D. Both ellipses intersect in the points with x = z = 0. This is how you get the cage.

Hmm, would this make it easier to derive cages for more than four points?
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 29, 2014 2:01 am

Secret wrote:ICN5D
Love those gifs


Thanks! Super cool, aren't they? Nothing better than live action footage of what we've been talking about.

Now I can double check my visualisation progress and see if there are errors in my drawings


Let me know if you want to see something specific, that I haven't made already. I'll totally make one on special request, for anyone. I'm hoping that the GIFs will stir up some more interest, and get others to ask questions. It's mostly been Marek and myself going back and forth, developing on new ideas. A really good one was the " Inflation Sequence " of ' A along B ' method of reducing them. We developed it based on Polyhedron Dude's naming system of 'small shape along large shape'. Come to find out, it's extremely useful for a visual aid, and the way it matches up in toratope notation is paramount to one's comprehension. We can now describe +10D toratopes very easily with this reduction.


I am miles behind this thread now due to uni and stuff, it might took a while to catch up and start makings cuts with you guys


I know, right? It evolved so very quickly. Well, again, feel free to ask about previous stuff. There's valuable insights scattered randomly all throughout this thread. It's tough to pick out the noteworthy ones, in the midst of everything else. I'm just an accomplished apprentice, Marek will always be the Zen Toratope Master :)




Marek wrote:Hmm, would this make it easier to derive cages for more than four points?



I'm thinking so. Ultimately, any and all margins come from the implicit surface equation, we just remove the final minor diameter to deflate it. In the way I see it, if tiger is :

((II)(II)) -- (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0

then a duoring would be :

(sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 = 0

and we can use :

(sqrt(x^2 + (y*sqrt(2)/2 + w*sqrt(2)/2)^2) - R1a)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - w*sqrt(2)/2)^2) - R1b)^2 = 0

to rotate 45 degrees, then cut to 3D:

(sqrt(x^2 + (y*sqrt(2)/2 + a*sqrt(2)/2)^2) - R1a)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - a*sqrt(2)/2)^2) - R1b)^2 = 0

where when a=0 , we have the infinitely thin tiger cage. The minor diameter R2 needs some value to flesh it out in CalcPlot, at least 0.25 with 40 cube rendering.




So, then for a duoring torus, the edge of a cyltorinder, we use the function from (((II)I)(II)) :

(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3^2 = 0

deflate R3 to make just the edge of a ((II)I)(II) , tracing out to 8 points in a 4x2 array:

(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 = 0

From this, we'll have to reconcile the property of the single major diameter, and the two equal mediums for a proper cage. The mediums R2a and R2b will be identical to the tiger cage rewriting, but I'm not sure how to add in the major R1. When it happens, we get close to the hexatangent cut of (((II)I)(II)), which has a tangent cluster that can inscribe an octahedron :o I'll have to play with the sliders, and get some values to reverse derive a technique. A good place to start may be:

(sqrt((sqrt(x^2 + y^2) - R1)^2 + (z*sqrt(2)/2 + v*sqrt(2)/2)^2) - R2a)^2 + (sqrt(w^2 + (z*sqrt(2)/2 - v*sqrt(2)/2)^2) - R2b)^2 = 0

From here, I suppose a rotation around XW or YW would be worthy to consider. Like, maybe:

(sqrt((sqrt(x^2 + (y*sqrt(2)/2 + w*sqrt(2)/2)^2) - R1)^2 + (z*sqrt(2)/2 + v*sqrt(2)/2)^2) - R2a)^2 + (sqrt((y*sqrt(2)/2 - w*sqrt(2)/2)^2 + (z*sqrt(2)/2 - v*sqrt(2)/2)^2) - R2b)^2 = 0

???




One can do this with the quad tiger cage from (((II)I)((II)I)), as it's two identical duorings, a smaller inflating a larger:

(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

and R3 deflates to :

(sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 = 0

then we rewrite to :

(sqrt((sqrt(x^2 + (y*sqrt(2)/2 + v*sqrt(2)/2)^2) - R1a)^2 + (z*sqrt(2)/2 + u*sqrt(2)/2)^2) -R2a)^2 + (sqrt((sqrt(w^2 + (y*sqrt(2)/2 - v*sqrt(2)/2)^2) - R1b)^2 + (z*sqrt(2)/2 - u*sqrt(2)/2)^2) - R2b)^2 = 0

for a true 45x45 degree slice of a di-duoring, duoring inflated duoring. But, since this structure is a multiplication of the surface of two torii, which have a big+small circle, we get a binomial expansion into big+big , big+small , small+big , and small+small . Which cuts at 45 degrees to make only two tiger cages of big+big and small+small .





This inspires me to make a showcase of just the polyrings themselves. I already have a few pics and GIFs made just for it, too. It'll be good to discuss the edges of the open toratope prisms, as all closed toratopes are nothing more than an inflated edge. I should probably make another thread on opens as well, with their surtopes and enumeration algorithm.


Which brings me back to a previous post, about the open toratope implicit equations:

Implicit equation for open toratopes are easy -- you just don't use all coordinates. For example, x^2 + y^2 = 1 is equation of circle in 2D, but equation of infinite cylinder in 3D and equation of circle x plane in 4D.


How can I close up the ends on a cylinder, in the 3D rendering program? I might want to make slice animations of a cubinder, or cyltorinder ( duocylinder torus ). Also, what would the equation for a cyltrianglinder be? It has three cylinders joined by a triangle torus in a triangular configuration. In my notation it'll be :

[ I>IO • {3^IOI + I>(O)} • {3:IO + 3^I(O)} • 3:(O) •-• n ]

which I'm inclined to develop a way to convert it into implicits. That'll be the next big step!
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 29, 2014 2:53 am

wendy wrote:Swirlprism is in essence a transform from a 3d space into 4d cliffords. So if you make a icosahedron edgeframe and make it a swirl prism, you would get a genus 11 tiger.


That would be cool to see! So, are you thinking it would cut down to 44 circles, or something like that?
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Re: The Tiger Explained

Postby Marek14 » Sun Jun 29, 2014 5:27 am

ICN5D: I didn't mean cages for higher toratopes, though, I meant for example a cage based on six points.

What we need is something with one cut equal to three vertical circles constructed on three sides of hexagon and second cut equal to three vertical circles constructed on the other three sides...
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 29, 2014 10:10 pm

Oh, that's right, the six legged beasts. I did check out that function you posted. Interesting to see the tiger cage expand into a 4-way tunnel, then split into four torii. I see the relationship with a tiger and a square, and how it relates to the 2D cut of a duocylinder. The vertices of the square cut is the 2x2 trace of the margin.

Hmmm, maybe if by starting with a hexagon, then extruding into 3D to make a hexagon prism, then lathing into 4D to make a " cylhexagoninder " with a 2D cut of a hexagon will be a good place to start from? This 4D cylindrical shape will have six cylinders joined by a hexagon torus as the N-1 surtopes, all bound by an N-2 margin that might inflate to a six-legged beast.
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 30, 2014 4:16 am

Here we go:


9D
((((II)I)(II))(II)(II)) - 1x [21-torus 20-tiger 220-triger] , sphere --> trioring --> duoring --> ring , ((((maj)sec1)(sec2)tert)(quat1)(quat2)min)
------------------------------------------------------------------------------------------------------------------------------------------------
8D
((((I)I)(II))(II)(II)) - 2x [220-tiger 220-triger] (((II)(II))(II)(II)) stacked 2 along a line in major1 dimension
((((II))(II))(II)(II)) - 2x [220-tiger 220-triger] (((II)(II))(II)(II)) as a concentric major1 pair
((((II)I)(I))(II)(II)) - 2x [211-ditorus 220-triger] ((((II)I)I)(II)(II)) stacked in a 1x1x1x2x1x1x1x1 tertiary1 column
((((II)I)(II))(I)(II)) - 2x [21-torus 20-tiger 21-tiger] ((((II)I)(II))(II)I) stacked in a 1x1x1x1x1x1x1x2 minor column
--------------------------------------------------------------------------------------------------------------------------------
7D
(((()I)(II))(II)(II)) - empty
((((I))(II))(II)(II)) - 4x [21-torus 220-triger] (((II)I)(II)(II)) stacked in a 1x1x4x1x1x1x1 medium1 column
((((I)I)(I))(II)(II)) - 4x [21-torus 220-triger] (((II)I)(II)(II)) stacked in a 2x1x2x1x1x1x1 major/med1 square array
((((I)I)(II))(I)(II)) - 4x [220-tiger 21-tiger] (((II)(II))(II)I) stacked in a 2x1x1x1x1x1x2 maj1/minor square array
((((II))(II))(I)(II)) - 4x [220-tiger 21-tiger] (((II)(II))(II)I) as concentric major1 pair stacked in a 1x1x1x1x1x1x2 minor column
((((II))(I))(II)(II)) - 4x [21-torus 220-triger] (((II)I)(II)(II)) as concentric major pair stacked in a 1x1x2x1x1x1x1 medium1 column
((((II)I)())(II)(II)) - empty
((((II)I)(I))(I)(II)) - 4x [211-ditorus 21-tiger] ((((II)I)I)(II)I) stacked in a 1x1x1x2x1x1x2 tertiary1/minor square array
((((II)I)(II))()(II)) - empty
((((II)I)(II))(I)(I)) - 4x [21-torus 20-tiger 2-torus] ((((II)I)(II))II) stacked in a 1x1x1x1x1x2x2 minor square array
-----------------------------------------------------------------------------------------------------------------------------------
6D
(((())(II))(II)(II)) - empty
(((()I)(I))(II)(II)) - empty
(((()I)(II))(I)(II)) - empty
((((I))(I))(II)(II)) - 8x trigers ((II)(II)(II)) stacked in a 4x2 maj1 flat rectangle array
((((I))(II))(I)(II)) - 8x [21-torus 21-tiger] (((II)I)(II)I) stacked in a 1x1x4x1x1x2 med1/minor rectangle array
((((I)I)())(II)(II)) - empty
((((I)I)(I))(I)(II)) - 8x [21-torus 21-tiger] (((II)I)(II)I) stacked in a 2x1x2x1x1x2 maj/med1/minor cube array
((((I)I)(II))()(II)) - empty
((((I)I)(II))(I)(I)) - 8x [220-tiger 2-torus] (((II)(II))II) stacked in a 2x1x1x1x2x2 maj1/minor cube array
((((II))())(II)(II)) - empty
((((II))(I))(I)(II)) - 8x [21-torus 21-tiger] (((II)I)(II)I) as concentric major pairs stacked in a 1x1x2x2x1x1 med1/minor square array
((((II))(II))()(II)) - empty
((((II))(II))(I)(I)) - 8x [220-tiger 2-torus] (((II)(II))II) as concentric major1 pairs stacked in a 1x1x1x1x2x2 minor square array
((((II)I)())(I)(II)) - empty
((((II)I)(I))()(II)) - empty
((((II)I)(I))(I)(I)) - 8x [2112-tritorus] ((((II)I)I)II) stacked in a 1x1x1x2x2x2 tertiary/minor cube array
((((II)I)(II))()(I)) - empty
----------------------------------------------------------------------------------------------------------------------------------
5D
(((())(I))(II)(II)) - empty
(((())(II))(I)(II)) - empty
(((()I)())(II)(II)) - empty
(((()I)(I))(I)(II)) - empty
(((()I)(II))()(II)) - empty
(((()I)(II))(I)(I)) - empty
((((I))())(II)(II)) - empty
((((I))(I))(II)(I)) - 16x [221-tiger] ((II)(II)I) stacked in a 4x2x1x1x2 maj1/min cuboid array
((((I))(II))()(II)) - empty
((((I))(II))(I)(I)) - 16x [212-ditorus] (((II)I)II) stacked in a 1x1x4x2x2 med/min cuboid array
((((I)I)())(I)(II)) - empty
((((I)I)(I))()(II)) - empty
((((I)I)(I))(I)(I)) - 16x [212-ditorus] (((II)I)II) stacked in a 2x1x2x2x2 maj/med/min tesseract array
((((I)I)(II))()(I)) - empty
((((II))())(I)(II)) - empty
((((II))(I))()(II)) - empty
((((II))(I))(I)(I)) - 16x [212-ditorus] (((II)I)II) as concentric major pairs stacked in a 1x1x2x2x2 med/min cube array
((((II))(II))()(I)) - empty
((((II)I)())()(II)) - empty
((((II)I)())(I)(I)) - empty
((((II)I)(I))()(I)) - empty
((((II)I)(II))()()) - empty
------------------------------------------------------------------------------------------------------------------------------------
4D
(((())())(II)(II)) - empty
(((())(I))(I)(II)) - empty
(((())(II))()(II)) - empty
(((())(II))(I)(I)) - empty
(((())(I))(II)(I)) - empty
(((()I)())(I)(II)) - empty
(((()I)(I))()(II)) - empty
(((()I)(I))(I)(I)) - empty
(((()I)(II))()(I)) - empty
((((I))())(I)(II)) - empty
((((I))(I))(I)(I)) - 32x spheritoruses ((II)II) stacked in a 4x2x2x2 maj/min array
((((I))())(II)(I)) - empty
((((I))(I))(II)()) - empty
((((I))(II))()(I)) - empty
((((I)I)())()(II)) - empty
((((I)I)())(I)(I)) - empty
((((I))(I))()(II)) - empty
((((I)I)(I))()(I)) - empty
((((I)I)(II))()()) - empty
((((II))())()(II)) - empty
((((II))())(I)(I)) - empty
((((II))())()(II)) - empty
((((II))(I))()(I)) - empty
((((II))(II))()()) - empty
((((II)I)())()(I)) - empty
((((II)I)(I))()()) - empty
-------------------------------------------------------
3D
All Empty !




I don't think I've ever parsed every cut like this before. At least, not for a 9D toratope. It's a good exercise, for sure!
Last edited by ICN5D on Mon Jun 30, 2014 4:53 am, edited 1 time in total.
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