Johannes Kepler named the compound of two tetrahedra the “stella octangula,” thus helping make it one of the best-known polyhedral compounds today. This variant uses triakis tetrahedra in place of the Platonic tetrahedra in that compound. The triakis tetrahedron is a Catalan solid, and is dual to the truncated tetrahedron.
Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.
Some polyhedral compounds are well-known, such as the compound of five cubes, while others are less famous. I had never heard of this compound before building one today (virtually, not as a physical model). However, a quick Google-search told me that I was not the first person to discover it.
Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.
Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.
This is like a tessellation, but in three dimensions, rather than two. The pattern can be repeated to fill all of space, using cubes (yellow), truncated octahedra (blue), and great rhombcuboctahedra, also known as truncated cuboctahedra (red).
Software credit: see www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.
This was made by the augmentation of an icosahedron, using snub dodecahedra on each of its twenty faces. I used software available at http://www.software3d.com/stella.php.
There are only a few polyhedra which can fill space without leaving gaps, without “help” from a second polyhedron. This filling of space is the three-dimensional version of tessellating a plane. Among those that can do this are the cube, the truncated octahedron, and the rhombic dodecahedron.
If multiple polyhedra are allowed in a space-filling pattern, this opens new possibilities. Here is one: the filling of space by cuboctahedra and octahedra. There are others, and they are likely to appear as future blog-posts here.
Software credit: I made this virtual model using Stella 4d, polyhedral-manipulation software you can buy, or try as a free trial download, at http://www.software3d.com/Stella.php.
This was made by the augmentation of a great dodecahedron, using snub dodecahedra on each face. I used software available at http://www.software3d.com/stella.php.
In this construction, I took the polyhedral cluster found in the last two posts, and augmented every pentagonal face with yet another great dodecahedron. I used software you can find at http://www.software3d.com/stella.php.
To make this, I used the polyhedron found in the last post, using software I bought at http://www.software3d.com/stella.php, to alter the coloring scheme to make each face a different color unless it is parallel to another, in which case it has the same color as all parallel faces.
Which coloring-scheme do you like best?
Begin with one great dodecahedron, and then augment each face with another, and you get this. I used software you may find at http://www.software3d.com/stella.php to make it.
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