Noble polyhedra. Where can I find them?

For general discussion of polyhedra, not necessarily Stella-specific.
senkoquartz
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Joined: Wed May 20, 2020 1:41 pm

Re: Noble polyhedra. Where can I find them?

Post by senkoquartz » Wed Aug 05, 2020 1:29 pm

Returning to this topic, having to sift through Bruckner's catalogue to find non-compounds and non-repeats initially left we with a disinterest for noble compounds, but now I'm getting interested in them.

I'm surprised to learn:

disphenoids can be self-dual,
self-dual disphenoids can form noble compounds which are non-self-dual, which is very counterintuitive,
the uniform compounds presipsido/passipsido, presipsi/passipsi, snu, and dis are all noble, despite not being regular

On Bruckner's models:
- Duals (22,3)/(22,4) lose their cuboctahedral/rhombicdodecahedral arrangement (12verts/12faceplanes) when distorted, and if you deform them without breaking apart components at the vertices/faceplanes, they can retain isogonality/isohedrality respectively but lose their octahedral symmetry, becoming just tetrahedral. That also goes for the other facetings/stellations for cuboctahedron and its dual.
- When disphenoid compounds with 48 verts/faces (like girco/dual) double up the verts/faceplanes this way, they take on the arrangement of tic/dual and toe/dual. But there is not a special set of proportions needed to do this.
- Bruckner's degenerate non-compound nobles (where faces revisit their vertices) are derived from cases of the above
- (22,14), (23,10), (24,4), and (24,8) are both self-duals and made of disphenoids. (24,8) is self-dual, made of crown polyhedra.
- Duals (24,7)/(26,8) have vert/face arrangement of icosidodecahedron/rhombic triacontahedron, will lose it like (22,3)/(22,4) do when distorted, and can't retain their respective vert/face transitivity at all while also keeping 30 verts/faceplanes. All other triacontal facetings/duals are rigid like this too.
- Duals (26,9)/(26,7) also have this triacontal rigidity
- Duals (25,1)/(25,5) have it too, and you can remove every other cell to get another, chiral, noble pair.
- Duals (25,2)/(21,14) are a special case of both vertices and face planes doubling up.
- The degen noble in (25,11) and (8,10) seems to have faces built out of congruent scalene triangles, giving a strange echo of the other degen nobles' structures.
- (25,12)(half) has the same convex hull as the compound of 4 triangular prisms
- Duals (27,4)/unlisted and (27,5)/unlisted come close to doubly lining up like (25,2)/(21,14), but don't.

senkoquartz
Posts: 23
Joined: Wed May 20, 2020 1:41 pm

Re: Noble polyhedra. Where can I find them?

Post by senkoquartz » Wed Aug 05, 2020 2:00 pm

Also, regarding rigidly symmetrical polyhedra -

There are only three shapes I know of that are both rigidly isogonal AND rigidly isohedral, and only one is a non-compound.

The Bruckner catalogue's Stellated Icosahedron B is the non-compound. Because it's both a dodecahedron faceting and an icosahedron stellation, it's rigid-isogonal and rigid-isohedral, respectively. This is also the only non-regular noble polyhedron with isogonal faces. It's also the simplest non-regular noble in terms of vert/edge/face count: 20/60/20.

The compounds are the 5 tetrahedra and 10 tetrahedra, for the same reason - dodec verts and icosa faces. But what really confuses me is that the 10 tetrahedra compound has 40 faces total, and yet appears to be completely rigid. Normally when an icosahedron/rhombictriacontahedron stellation has compound faces, they can be tilted out of sharing the same plane, and the polyhedron will remain face-transitive - it'll just have 60 face planes instead of 20 or 30. But I don't understand how one could tilt the compound faces of the 10 tetrahedra without taking away the face-transitivity all together. 40 rigidly transitive faces is weird.

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marcelteun
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Re: Noble polyhedra. Where can I find them?

Post by marcelteun » Mon Aug 17, 2020 8:49 am

Ulrich wrote:
Thu Jul 02, 2020 6:58 am
My paper about exploring noble polyhedra with Stella4D is now online at the Bridges Website:

http://archive.bridgesmathart.org/2020/ ... 20-257.pdf

Ulrich
Pity the Corona caused the event to be cancelled. After years of skipping the event, I was planning to come this year. Congrats with your article! Thanks for sharing!

Marcel

ps sorry, I missed some posts.

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