If you tidy-facet the 24-vert Archimedean solids, with the "Must Be Spikey" requirement checked in the faceting menu, some of the closer-to-the-end results have duals made of isohedral heptagrams, classically starry-looking.

See thread title.

I want to make compounds of chiral shapes and their reflections

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 https://i.pos...

I quickly made a folder of the triacontahedra with singular, non-compound faces. https://1drv.ms/u/s!AgVMslM5Ti8XjQ-wg6MtewMmH8yT?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 li...

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...

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 ...

Microsoft OneDrive link to files: https://1drv.ms/u/s!AgVMslM5Ti8XjQ52hN8Cqx96Qu6-?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....

Yeah, IMO, I think stella octangula would totally be the best choice for knight. A cuboctahedron clashes because all the others are regular. A stella octangula is regular, and also fits because pawns/tetrahedra are the smallest/simplest pieces, kings+queens/dodeca+icosa are the most important/comple...

Knight could be a tetrahemihexahedron maybe? Or a stella octangula. As for king vs. queen and icosa vs. dodeca, a rook and cube are sort of more masculine than a bishop and octahedron, so that would carry over to rook->king and bishop->queen, and dodeca corresponds to cube while icosa goes with octa...

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 icosahedr...

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 comp...

but alas, the thread title should say "facetings where the DUAL'S face's vertices lie on its edges"

I was searching for isohedral triacontahedra by faceting the icosidodecahedron and checking its duals. But 95% of them end up having faces with verts lying on their own edges. Is there a way to skip over these?

*regarding that last question - i mean when I go through the 227 rhombic triacontahedron stellations for example, they outwardly look like isohedra but the faces are broken up

For the past day I've been interested in rigidly symmetrical polyhedra in general. (I'll be ignoring infinite dihedral families here.) There are countless types of isogonal polyhedra (and their isohedral duals), but the ones that are rigid and lose their transitivity when distorted seem more unique ...