A Polyhedral Investigation, Starting with an Augmentation of the Truncated Octahedron

If one starts with a central truncated octahedron, leaves its six square faces untouched, and augments its eight hexagonal faces with trianglular cupolae, this is the result.

Seeing this, I did a quick check of its dual, and found it quite interesting:

After seeing this dual, I next created its convex hull.

After seeing this convex hull, I next creating its dual:  one of several 48-faced polyhedra I have found with two different sets of twenty-four kites as faces, one set in six panels of four kites each, and the other set consisting of eight sets of three kites each. I think of these recurring 48-kite-faced polyhedra as polyhedral expressions of a simple fact of arithmetic: (6)(4) = (8)(3) = 24.

I use Stella 4d (available at http://www.software3d.com/Stella.php) to perform these polyhedral transformations. The last one I created in this particular “polyhedral journey” is shown below — but, unfortunately, I cannot recall exactly what I did, to which of the above polyhedra, to create it.

I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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2 Responses to A Polyhedral Investigation, Starting with an Augmentation of the Truncated Octahedron

1. Dave Smith says:

I have just come across these models by mistake – they look great! Did you make these yourself? I notice that one uses a bow-tie trapezoid – does it have the same proportions as the one I sent you?

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2. I made them using Stella 4d, yes. You’ll actually find a lot of bow-tie polyhedra on various posts here. These trapezoids, however, don’t have the same angles as in your discoveries.

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