Comparing Regular-Faced Toroidal Polyhedra 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.
 A: 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).
A: 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
A: 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).

.............................
