Compound of 4 tridiminished icosahedra

[quickfur]

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

[Klitzing]

nice observation, quickfur!
however, this compound will have a chiral tetrahedral symmetry only.
--- rk

[quickfur]

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.

[Mecejide]

This compound of four tridiminished icosahedra is also the vertex figure of a degenerate uniform compound of five snub 24-cells with the same cells as a 600-cell plus an icosahedral 120-cell. It has what Bowers calls ixitic symmetry.

[quickfur]

Nice!

[mr_e_man]

Divide the vertices of the 600-cell into 5 groups of 24, delete one group to make the snub 24-cell, and look at one of the resulting icosahedra.
The four teddi's are the possible results of deleting a second group of 24 vertices to make bidex.