The Compound of the Truncated Isocahedron and the Pentakis Dodecahedron, with Related Polyhedra

The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.

Pentakis dodecahedron and truncated icosahedron

The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.

Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.

dual of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Next, here is the compound of the last two polyhedra shown.

dual and base compound of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Continuing this process, here is the convex hull of the compound shown immediately above.

Convex hull

This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.

Dual of Convex hull

This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.

All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php

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