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.

AUGMENTED TRUNCTAED OCTAHEDRON

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

DUAL OF AUGMENTED TRUNCATED OCTAHEDRON

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

Convex hull x

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.

48 KITES AGAiN

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.

Convex hull OF AUGMENTED CUBOCTAHEDRON

2 thoughts on “A Polyhedral Investigation, Starting with an Augmentation of the Truncated Octahedron

  1. 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?

    Liked by 1 person

  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.

    Like

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