Comparing Regular-Faced Toroidal Polyhedra

Question

Many apologies ahead of time, I have no idea how to phrase this question, and I'm certainly way out of my element. I'll do my best but please go easy on me.

I wanted to make a polyhedra that was in the shape of a doughnut. For whatever reason, I thought it would be cool if that polyhedra had regular-polygon faces. With some online-research, I found these things called Stewart Toroids that were seemingly what I was looking for. The problem was that they all looked... well... ugly? I think Stewart had some rules for how he made his toroids and maybe that had something to do with why they didn't really look much like a doughnut to me.

Anyway I set about making my own, and I made something that looked more doughnut-like. The driving rule I used to make it was to try and make every vertex (where faces met) as smooth as possible, which I interpreted to mean I needed to minimize the angle change between all the faces. I think for that exact characteristic (least angle change between faces) you can't do better than this shape does (worst angle change is 36 deg)

(sorry, I know those pictures are bad)

So I guess my question is if any of that makes sense.

• Is it okay thinking one polyhedra could be more "torus" than another?
• Is there a smart way to measure that, or is this purely a subjective thing?
• Does the idea of smoothing out a polyhedra actually help or is that really more cosmetic?
• Is minimizing the angle between adjacent faces the right way to maximize smoothness?

EDIT: It seems I've attracted a number of people also interested in making more such polyhedra, which is fine, I certainly had fun making this thing. But the question is not "can you make these polyhedra", it is "how can we compare them?"

There's nothing stopping me from making a torus with a million little square faces, approximating a torus in the same fashion a bunch of pixels can approximate a circle. The thing I want to know is if math tells us how similar two shapes are, such that there can be an official method for comparing two polyhedral doughnuts.

Answer 1

I am not sure if this helps.

To construct a torus, you have an area of negative curvature (the inside), and an area of positive curvature (the outside).

The net curvature of a torus is 0.

At each vertex we can sum the angles and subtract from $360$ (or $2\pi$) to find that vertices contribution to the total curvature.

you need a total contribution of $720^\circ$ from the vertices on the outside, and $-720^\circ$ contribution from the vertices on the inside.

If we used only equilateral triangles, we need 12 vertices with 5 triangles at these vertices, and 12 vertices with 7 triangles at the vertex, (and some number of 6 triangles at each vertex).

Answer 2

Your volume can be described as the assembly of 6 so-called (small) rhombicosidodecahedra (see Figure below) "stitched" together by decagonal-based prismatic shapes

(copied from the site http://www.mi.sanu.ac.rs/vismath/zefiro1/_icosahedral_polyhedra_2007_11_15.htm ; in their fig. 3, one finds a certain generation mode).

I think we have there a way to construct other shapes using other Archimedean polyhedra, the stitching mode being in some cases a little different, for example by using antiprisms (https://en.wikipedia.org/wiki/Antiprism) instead of prismatic shapes.

An article connecting Archimedean solids and in particular rhombicosidodecahedra to quaternions (see its fig. 10) : https://arxiv.org/ftp/arxiv/papers/0908/0908.3272.pdf

Answer 3

42 sided Goldberg polyhedron made with Zometool parts. For strength, there is a central ball, spokes out to an icosahedron (blue), then many (yellow) spokes out to the polyhedron itself (pentagons in blue, then hexagons with some blue and some red sides).

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