Some Ten-Part Polyhedral Compounds

While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.

Faceted Dodeca and compound of ten elongated octahedra.gif

Here’s what this compound looks like with the edges and vertices hidden:

Faceted Dodeca and compound of ten elongated octahedra without edges and vertices.gif

Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.

Faceted Dodeca one part of ten with edges and vertices.gif

Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.

Faceted Dodeca one part of ten.gif

I made all these polyhedral transformations using Stella 4d, a program you can try for yourself at this website. Stella includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).

The next thing I wanted to try was to make the octahedra regular. Stella has a function for that, too, and here’s the result: a compound of ten regular octahedra.

compound of ten regular octahedra.gif

My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.

compound of ten cubes.gif

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