A Non-Convex “Cousin” of the Cuboctahedron

appears to be a facted cuboctahedron

My guess is that this is a faceting of the cuboctahedron, but I didn’t use faceting when I made it with Stella 4d (a program you can try here), so I am not sure about this. Based on its appearance, however, it is clearly related, in some manner, to the cuboctahedron, for the cuboctahedron is its convex hull.

A Cuboctahedral Cluster of Rhombic Dodecahedra

cuboctahedron of Rhombic Dodeca

It is well-known that the cuboctahedron and the rhombic dodecahedron are dual polyhedra. However, until I stumbled upon this, I was unaware that rhombic dodecahedra could actually be arranged into a cluster with the overall shape of a cuboctahedron.

[Software credit: see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]

Cuboctahedral Cluster of Rhombic Triacontahedra

Augmented Rhombic Triaconta

Due to their high number of planes of symmetry, rhombic triacontahedra make excellent building blocks to build other polyhedra. To make this, I used a program called Stella 4d, which you can try right here.

A Collection of Four Polyhedra Decorated with Mandalas

First, a cuboctahedron.

Rotating Cubocta with rotating mandalasNext, its dual, the rhombic dodecahedron.

Rotating RD with rotating mandalas

And, after that, the icosidodecahedron.

Rotating Icosidodeca with rotating mandalas

And finally, its dual, the rhombic triacontahedron.

Rotating RTC with rotating mandalas

All of these rotating images were assembled using Stella 4d, available at http://www.software3d.com/Stella.php.

Dodecahedral Cluster of Cuboctahedra and Icosidodecahedra

Augmented IcosidodDSJFGSca

I made this using Stella 4d:  Polyhedron Navigator, software you may try for yourself at http://www.software3d.com/Stella.php.

Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

Image

Pulsating Cuboctahedron, Featuring Enneagrammic Mandalas

The enneagramic mandalas on the square faces of this cuboctahedron are from the last post, with inverted-color, smaller versions of the same image on the triangular faces. These mandalas were created using Geometer’s Sketchpad and MS-Paint. Projecting them onto the faces of the cuboctahedron, and then creating this rotating, pulsating .gif image, however, took a third program: Stella 4d, which you can buy, or try for free, at http://www.software3d.com/Stella.php.