A list of every* isohedral triacontahedron, plus every* rigidly isohedral polyhedron, and a list of nobles

[senkoquartz]

Microsoft OneDrive link to files: https://1drv.ms/u/s!AgVMslM5Ti8XjQ52hN8 ... -?e=nll1PQ

If you go through "Tidy Facetings and Duals" of the icosidodecahedron, there are 4379 results, 3671 of which have "edgevert" faces (vertices hitting their own edges). That leaves 708 ones with "proper" faces. Including the "edgeverts" leads to the majority of the list being a slog of ugly, repetitive faces. I personally find maybe 1% of the edgevert ones to be interesting enough look at twice, whereas I like 100% of the proper ones. I spent __ hours sorting through every result to find and compile these "proper" ones.

Rhombic triacontahedron stellations and icosidodecahedron facets are "rigid" as explained below, the is a catalogue of every* proper one, as well as four other catalogues of "rigid" shapes: icosahedron/dodecahedron stellations/facets, cube/octahedron stellations/facets, and noble polyhedra.

*When I say "every", it does not include duals of icosidodecahedron facets with faces passing through the center, of which there are many, because I was interested in the isohedral duals and they are infinite "crons" in those cases, not fun. The process I used also doesn't create 30hedra with compound faces that have more than 4 polygons in the same plane - off the top of my head the "04" triacontahedron in my list could be seen as having sextuple-compound faces. And as stated, it doesn't include edgeverts.

I will reply with an explanation of what I mean by "rigid".

[senkoquartz]

This following is only referring to polyhedra with greater than dihedral symmetry:

24face/24vert polyhedra can be distorted and retain their isohedrality/isogonality, as can 60face/60vert and 120face/120vert. Isohedrality/isogonality of order 4/6/8/12/20/30 is more rigid.

The tetrahedron, cube, and octahedron can be squashed/stretched and remain isohedral or isogonal (or both, for tetrahedron), but get downgraded to dihedral symmetry.

The cuboctahedron / rhombic dodecahedron can be distorted and retain isogonality / isohedrality respectively, but get downgraded to tetrahedral symmetry and lose isotoxality (edge-transivity).

Dodecahedron vert arrangement is rigidly isogonal, and thus the dual icosahedron face arrangement is rigidly isohedral. (doe face arrangement is flexibly transitive, as is ike vert arrangement, but distorting will downgrade them to tetrahedral symmetry). This means that dodecahedron facets and the corresponding icosahedron stellations are rigid in the same way.

Icosidodecahedron is rigidly isogonal, rhombic triacontahedron is rigidly isohedral, same thing as above applies regarding facetings and stellations.

20hedra and 30hedra with compound faces can be distorted, but face planes will break apart and become 60 instead of 20/30. Same goes for the 20vert/30vert duals.

Noble polyhedra are generally flexibly vert and face transitive, but only rigidly both at the same time and thus remaining noble.

The compounds of: 5 cubes, 5 octahedra, 5 tetrahedra, and 10 tetrahedra are both rigidly face- and vert- transitive. One noble polyhedron that is both an ike-stellation and doe-facet also has this property. I don't know if noble id-facets and their duals are like this too. The 5 cube compound is both a doe-facet and rhombictriacontahedron-stellation, and the 5 octahedra compound is both an ike-stellation and id-facet.

[senkoquartz]

Oh, I forgot to explain the filename notation for the 708 triacontahedra:

filename examples:

05 H
04 L b
47 P JF
44 H DD compound

The numbers at the beginning designate the external appearance of the polyhedron. The proceeding letter refers to the number of sides per face (D is 4 sides, H is 8, L is 12, V is 22, etc). If there are a group of upper-case letters after the first letter, it's a compound face and those are its components. If there is a lower case letter, it's differentiating between other entries that otherwise would have the same filename. Notes about the polyhedron like "compound" or "noble" or "quasiregular" are included at the end of the filename.

Also, for 708 list: red face in front means non-compound face, blue means compound, green means chiral compound face for a polyhedron that has an externally non-chiral appearance, pink is like green but non-compound faces.

[senkoquartz]

I quickly made a folder of the triacontahedra with singular, non-compound faces. https://1drv.ms/u/s!AgVMslM5Ti8XjQ-wg6M ... T?e=PWghFb

There are 288. Among them:

4 4gons
24 8gons
8 10gons
57 12gons
25 14gons
72 16gons
53 20gons
19 18gons
10 22gons
15 24gons
1 26gon

1 compound (5 cubes)

In the list with compound faces, there are 4 28faces, but none are non-compound

There is one more limitation of Stella's icosidodecahedron faceting, which is that it doesn't produce results with vertex valence >28. If there are any ≥30gon triacontahedra, I don't know about them

[senkoquartz]

At least one isohedron was missed - the third rhombic triacontahedron fully supported stellation, which has 8gon faces, very similar to the preceding stellation. The 5 cube compound also didn't show up, though I caught that one. So there could be others missing from my list, yes indeed


(missing 03 H DD compound)

[senkoquartz]

Two new uploads:
This is a list of every isohedral polyhedron and compound that loses T/O/I symmetry and/or loses isohedrality when distorted.
It includes all the ones with non-compound faces, plus a couple with compound faces that still lose symmetry when distorted.
It includes three new models: 1) the second rhombic triacontahedron stellation, which was simply missed before, 2) a compound of a chiral 30hedron and its mirror image, which has the same outward appearance as another, noble, 30hedron, and 3) another 30hedron mirror compound that also looks like another, noble, 30hedron.
https://1drv.ms/u/s!AgVMslM5Ti8XjzH3LKm ... k?e=S2KewX
Tetrahedron, cube, octahedron and stella octangula downgrade to dihedral symmetry when distorted
Rhombic dodecahedron stellations downgrade to tetrahedral symmetry when distorted
Dodecahedron stellations downgrade to tetrahedral symmetry when distorted
All others cannot be distorted without losing isohedrality
This list is for isohedral shapes, not isogonal ones, so it doesn't include isogonal hemipolyhedra and their cron duals.

And this list has a couple examples that the original 30plane isohedra list missed, and is not complete. There are three with 4 faces/plane which are fairly simple and elegant, two with 2 congruent faces/plane, ("missed 23 V KK" is actually a new outward appearance, not like '23') and one that is included for being very messy.
https://1drv.ms/u/s!AgVMslM5Ti8XkXFBYwp ... 4?e=h4Z3Ak

[Ulrich]

Great work! Thanks for sharing.
I can hardly stop myself from building a paper model of 198 42 Rb immediately.
Others like 309 93 L or 301 85 L would be a challenge.

Ulrich

[senkoquartz]

edit on icosahedron/rhomb-triacontahedron noble stellations:
If they have 60 vertices, 60-vert polyhedra are flexibly isogonal, so that would seem to mean these stellations are flexibly isogonal and rigidly isohedral, except 20/30 faced polyhedra can't be distorted while retaining icosahedral symmetry iirc. If a 20 or 30 faced noble polyhedron distorts and downgrades to tetrahedral symmetry, then its 60 vertices downgrade to tetrahedral symmetry, and you can't have 60 transitive vertices with only tetrahedral symmetry. So that makes those nobles both face- *and* vert- rigid. I think.

I'm exploring archimedean and uniform stellations. I want to catalogue isogonal stellations of uniforms, and I'm using faceting mode to create final stellations with internal structure of archimedeans/uniforms. The final stellation of id/ti/tid is a very funny and cute compound of a gissid, a final ike stellation, and a third shape. srid maxes out to a lovely extravagant spikey star made of 12 20gons, 30 24gons, and 20 24gons. sirco does a more modest cubic version. sird maxes out to something similar, but with only 2 kinds of faces.
More generally I want to know about the more neat and important isohedral / -gonal polyhedra.