# Filling Space With Rhombic Dodecahedra

This is the cuboctahedron, one of the Archimedean solids. Its dual, shown below, is the rhombic dodecahedron.

The rhombic dodecahedron has a property which sets it apart from most other polyhedra: it can fill space with copies of itself, leaving no gaps. The next stage of such growth is shown below.

The next step is to add more rhombic dodecahedra on each face.

One more set added, and the edge-length of the cluster reaches four rhombic dodecahedra.

This could be continued without limit. As is does, the overall shape of the cluster becomes more and more shaped like a cuboctahedron, which is back where we started. You can easily see this in the convex hull of the last cluster.

All of these rotating images were created using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

# Filling Space with Cuboctahedra and Octahedra

To get started packing space with cuboctahedra and octahedra, I started with a single octahedron, then augmented its square faces with additional cuboctahedra.

Next, I augmented each triangular face with a blue octahedron.

Next, I augmented each square face with a cuboctahedron.

Next, I added still more cuboctahedra.

The next step was to augment the yellow triangular faces with blue octahedra.

This process may be continued without limit. I used a program called Stella 4d to make these models, and you can try this software yourself, for free, at this website.

# Packing Space with Great Rhombcuboctahedra and Octagonal Prisms

…And so on….

[Software credit: I made these images using Stella 4d, which you can try for free right here.]

# Octahedra and Truncated Cubes Can Fill Space Without Leaving Any Gaps

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.

# Octahedra and Cuboctahedra Can Fill Space Without Leaving Any Gaps

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

# Honeycomb Made of Cuboctahedra and Octahedra

This is the three-dimensional version of what is called a tessellation in two dimensions. It fills space, and can be continued in all directions.

Software used: Stella 4d, available here.

# A Space-Filling Lattice of Truncated Octahedra

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.

# Bowtie Cubes in a Polyhedral Honeycomb

This polyhedron has been described here as a “bowtie cube.” It is possible to augment its six dodecagonal faces with additional bowtie cubes. Also, the bowtie cube’s hexagonal faces may be augmented by truncated octahedra.

These two polyhedra “tessellate” space, together which square pyramidal bifrustrums, meeting in pairs, which fill the blue-and-green “holes” seen above. This last image shows more of the “honeycomb” produced after yet more of these same polyhedra have been added.

This pattern may be expanded into space without limit. I discovered it while playing with Stella 4d, software you may try for free at this website.

# A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, 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.

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

And now, more yellow cubes.

This 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.

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

The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

Now I’ll add more of the red truncated cubes.

At this point, more yellow cubes are needed.

The next polyhedra added will be pink rhombcuboctahedra.

And now, more of the blue octagonal prisms.

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