Toratope surfaces as 2D gluings / "parameterization"

Discussion of shapes with curves and holes in various dimensions.

Toratope surfaces as 2D gluings / "parameterization"

Postby hamish_todd » Mon May 14, 2018 2:09 pm

Hey folks, I just joined and have two questions for y'all.

I have seen lovely things on this forum including:
Image
((sqrt(y^2+z^2)-b)^2+(x-a)^2) * ((sqrt(((sqrt(3)x-y)/2)^2+z^2)-b)^2+((x+sqrt(3)y)/2+a)^2) * ((sqrt((-(sqrt(3)x+y)/2)^2+z^2)-b)^2+((x-sqrt(3)y)/2+a)^2) = c^6
And similarly the "tiger cage" in here viewtopic.php?f=24&t=1858
And the cube-like cut of the "triger" here viewtopic.php?f=24&t=801&start=390

These are of great interest to me because they are realizations of genus 3 and 4 surfaces in R3, and they are "nice"/"simple"/"smooth", maybe even "canonical" or "fundamental" . Does anyone have knowledge of how to create an implicit function for an arbitrary genus surface?

Contrast with
Image
From https://www.youtube.com/watch?v=cermfDnqQ5M / http://www.math.uni-tuebingen.de/ab/GeometrieWerkstatt/
Object is made in S3 by starting with the clifford torus and having some kind of geometric flow. Then stereographically projected. Generalizes horribly, their other genus-n surfaces are revolting.

Other question
It is known that a genus-n surface can be acquired topologically by "gluing", see below or this video https://www.youtube.com/watch?v=G1yyfPShgqw. Does anyone know how to map points onto these surfaces?
Image
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby Marek14 » Wed May 16, 2018 9:26 pm

Hm, hard to say. I know that I've been thinking of a hypothetical objects called "mantis" and "spider" that would form "tiger cage" with six or eight bars, but I never managed to nail their equations down...
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby hamish_todd » Fri May 18, 2018 1:03 pm

So, in case it helps anyone else, I have come up with a reasonably good solution to this. Warning: it might be hard to understand, good luck to you if you want to do this

Watch this video https://www.youtube.com/watch?v=cermfDnqQ5M it begins with a single surface inside a tetrahedron. That tetrahedron is actually in S3, with S3 tiled by 6*4 = 24 tetrahedra. Try to replace this shape with the 16 cell, which is 16 tetrahedra.

The nice thing about the (starting) surface is that it has 4 edges. It is easy to derive those edges. One example of a tetrahedron inside the 16-cell tiling of S3 is one with vertices at

(1,0,0,0)
(0,1,0,0)
(0,0,1,0)
(0,0,0,1)

...and to get the edges connecting any of these vertices, simply slerp from one to the other. And in fact, to get any point on the surface of our starting surface, all we have to do is consider two opposite edges of the tetrahedron, slerp some amount along both of them, and then slerp between those two resulting points.

This gives us a very versatile tool: we can make lots of surfaces by tiling space by tetrahedra and sticking these surfaces in them. We have a simple square parameterization of this starting surface, so any other surface we make is just about gluing squares together. I might even try to do it with a triangle.

Note that, as depicted in the video, to make the surface continuous at an edge E, we make two copies of the tetrahedron around E and must rotate one around E by 180 degrees. In fact the final shape in the video is made of 12 of these surfaces, not the 24 that fill up the actual space.

Note also that this is very similar to making a hyperbolic paraboloid as a ruled surface by connecting edges of a tetrahedron up with straight lines as in eg this
Image
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby wendy » Mon May 21, 2018 8:17 am

Welcome, Hamish.

The octagon in your first post, is a cell from the hyperbolic tiling {8,8}, octagons, eight at a corner. This is a fairly interesting group in its own right, the cell of the {8,8} breaks into sixteen triangles of {8,3}. I looked at your video, in the second post, and the surface is walking four of the six rings of the {3,3,4} 16choron, following the petrie polygon, (which is an octagon).

The figure with three holes in it (the first one), appears to be a representation of {3,3,4} as a subgroup of order 3 of the 24ch {3,4,3}, which has octagons. But there are a number of subgroups of order 3 there, so i should need to look at the diagrams closer before making any particular representation here.

But none the same, the figures look very impressing, and certainly something i have not seen before.

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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby hamish_todd » Tue May 22, 2018 11:01 am

Thank you Wendy! How do you mean that the first surface is a "representation" of the 24 cell?
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby wendy » Tue May 22, 2018 11:48 am

I watched your video again. The subgroup here is [[4,2,4]] in [3,3,4] The first is order 1.08 (dec 128) the second is 3,24 (dec 384), which comes from not using two orthogonal circles, the red and green ones.

This leaves for the 16ch, some 16 edges, and 16 squares. But there are only eight vertices, giving an euler characteristic of 1+g, 8,16,16,1+g.

Since the three pairs of orthogonal circles are the three pairs of opposite faces of the cube, it comes down to [2,4] in [3,4], ie 16 in 48.
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby hamish_todd » Wed May 23, 2018 1:01 pm

I'm not sure I understand the notation "1.08 (dec 128)". Is this still related to the 24-cell?
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby Marek14 » Wed May 23, 2018 1:15 pm

hamish_todd wrote:I'm not sure I understand the notation "1.08 (dec 128)". Is this still related to the 24-cell?


Wendy has a habit of using base 120 number system.
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Re: Toratope surfaces as 2D gluings / "parameterization"

Postby ICN5D » Mon Oct 01, 2018 7:34 pm

I've been working on a process to get equations for these higher genus surfaces, off and on. I made that 3-bar cage surface (in first pic) by using a generalization of bi-polar cassini ovals. As we can see here, it's a product of 2 disjoint circle equations on the LHS (with no radius param), with a single radius parameter defined over them on RHS, " b^4 " .

((x-a)^2 + y^2) * ((x+a)^2 + y^2) = b^4

So, what I did was take the product of 3 toruses (w/no minor radius param) positioned as the fence arrangement, with a single set of points defined over them as c^6 :

((sqrt(y^2+z^2)-b)^2+(x-a)^2) * ((sqrt(((sqrt(3)x-y)/2)^2+z^2)-b)^2+((x+sqrt(3)y)/2+a)^2) * ((sqrt((-(sqrt(3)x+y)/2)^2+z^2)-b)^2+((x-sqrt(3)y)/2+a)^2) = c^6

Copy-Paste this URL to see the CalcPlot3D graph of the above equation:
Code: Select all
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So I guess this is a 'tri-toroidal cassini surface' , which at some self-intersecting values of a,b,c (a=2 , b=5 , c=2.5 for this one), we get the genus-2 multitorus as this cage-like object. It's not a perfect fit, since a 2-D slice through the cage has 3 ellipses with 2 foci, instead of 3 circles with one. I do have a much more elegant equation that defines a product of 3 circles in the vertices of an equilateral triangle:

degree-6 polynomial with cube roots of 3 circles in triangular array

Desmos plot

This solution must come from the true polynomial of a 3-prong in 3 variables and 3 radius parameters. It has cube root properties when you cancel a variable and take the 2d slice through the 3 bars of the cage. Very interesting!

Going further, one could use self-intersecting tetra-toroidal, penta-toroidal, etc, surfaces to form more complex cage-like structures in 3D.

I have been working on a 4D extension of these, which Marek14 calls the Mantis. It's a 'tri-tigroidal cassini surface' , in 4 variables and 4 radius parameters. That's a product of 3 tigers (clifford toruses, actually) in a triangular array, with a single set of points over them.

I also added an extra parameter that aligns the slice of 6 toruses to only 3 (making them overlap in 3 pairs), by using a non-orthogonal plane of rotation for the tigers (one of the infinite varieties of 3-torus between tiger ((II)(II)) and 3-torus (((II)I)I) ).

The plots of this equation give some promising results, but it still may not be the mantis. It is, however, a true 3-bar cage structure with tiger-like properties, which still may be of interest (and can generalize further).

I need to update with pics and gifs of this thing, since it's the closest I've ever come to forming Mantis.
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