Compound of 4 tridiminished icosahedra

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Compound of 4 tridiminished icosahedra

Postby quickfur » Sat Aug 06, 2022 3:13 am

Today, I discovered an interesting compound of J63, the tridiminished icosahedron.

Take four J63's, centered on the origin, oriented according to the symmetries of a regular tetrahedron. Then all their triangular faces will be disjoint and form a closed polyhedral surface in the shape of an inscribing regular icosahedron. Furthermore, their pentagonal faces together form the non-convex great dodecahedron.

One might be tempted to express this relationship with the (pseudo-)equation:

o5o3x + x5o(5/2)o ≡ 4 * J63

:lol:
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Re: Compound of 4 tridiminished icosahedra

Postby Klitzing » Sat Aug 06, 2022 6:46 pm

nice observation, quickfur!
however, this compound will have a chiral tetrahedral symmetry only.
--- rk
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Re: Compound of 4 tridiminished icosahedra

Postby quickfur » Mon Aug 08, 2022 2:49 pm

You're right, it only has chiral tetrahedral symmetry.

It answered a long-time question I had, of how to partition an icosahedron's faces into 4 transitive subsets. I guess it could also be used to construct a partitioning of a dodecahedron's faces into 4 transitive subsets. Both would have (chiral) tetrahedral symmetry.
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