Octahedra and Truncated Cubes Can Fill Space Without Leaving Any Gaps

Augmented Trunc Cube

I created this using Stella 4d, which you can try for free right here. It’s much like a tessellation, but in three dimensions instead of two.

Space-Filling Truncated Octahedra in a Rhombic Dodecahedral Cluster

The truncated octahedron is well-known as the only Archimedean solid which can fill space, by itself, without leaving any gaps. The cluster below shows this, and has the overall shape of a rhombic dodecahedron.

Augmented Trunc Octa.gif

It’s easier to see the rhombic dodecahedral shape of this cluster when looking at its convex hull:

Convex hull.gif

Both images here were made using Stella 4d, which you can try for free right here.

A Space-Filling Lattice of Truncated Octahedra

augmented-trunc-octa

Truncated octahedra are among the special polyhedra which can fill space without leaving any gaps. There are others, as well. This image was created using Stella 4d, software you may try, for yourself, right here. There is a free “try it before you buy it” download available.

A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, Cubes, and Octagonal Prisms

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms

This image above has only one polyhedron-type hidden from view, in the center:  a red truncated cube. Next, more of this pattern I just found will be added.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 2

The next step will be to add another layer of blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 3And now, more yellow cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 4This was an accidental discovery I made, just messing around with Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 5

Next up, I’ll add a set of pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 6The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 7Now I’ll add more of the red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 8At this point, more yellow cubes are needed.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 9The next polyhedra added will be pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 10

And now, more of the blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 11

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.

A Space-Filling Pair of Polyhedra: The Cuboctahedron and the Octahedron

Image

A Space-Filling Pair of Polyhedra:  The Cuboctahedron and the Octahedron

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.