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.
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.
A cuboctahedron sits at the center of this rotating cluster, but you can’t see it, because each of its fourteen faces (six squares and eight equilateral triangles) has another cuboctahedron, of equal size, attached to it.
Software credit: visit http://www.software3d.com/stella.php to try (or buy) the polyhedral-manipulation software I used to make this virtual model.
I made this, using Stella 4d, by augmenting each decagonal face of the cluster in the previous post with a truncated dodecahedron. You can give this program a try yourself, for free, at http://www.software3d.com/stella.php.
I made this, using Stella 4d, by augmenting the thirty square faces of a great rhombicosidodecahedron with additional great rhombicosidodecahedra. The result has one of these polyhedra located in each position which corresponds to a vertex of an icosidodecahedron.
You can give this program a try yourself, for free, at http://www.software3d.com/stella.php.
I made this, using Stella 4d, by augmenting each face of an octahedron with a rhombicosidodecahedron. You can give this program a try yourself, for free, at http://www.software3d.com/stella.php.
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