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