Octahedra and Cuboctahedra Can Fill Space Without Leaving Any Gaps

Augmented Cubocta

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

Augmented Cubocta.gif

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.

Spectral Circles on a Cuboctahedron

Cubocta

I used three programs to make this: Geometer’s Sketchpad, MS-Paint, and Stella 4d. The third of these may be tried for free at http://www.software3d.com/Stella.php.

A Central Icosidodecahedron, Augmented with Twenty Cuboctahedra, and Twelve More Icosidodecahedra

Augmented Icosidodeca aug with 20 cuboctas and 12 icosidodecas color scheme two

Above and below, you will find two different coloring-schemes for this particular cluster of polyhedra. I made both of these rotating images using Stella 4d, software you can buy, or try for free, right here.

Augmented Icosidodeca aug with 20 cuboctas and 12 icosidodecas

Various Views of Three Different Polyhedral Compounds: Those of (1) Five Cuboctahedra, (2) Five of Its Dual, the Rhombic Dodecahedron, and (3) Ten Components — Five Each, of Both Polyhedra.

Polyhedral compounds differ in the amount of effort needed to understand their internal structure, as well as the way the compounds’ components are assembled, relative to each other. This compound, the compound of five cuboctahedra, and those related to it, offer challenges not offered by all polyhedral compounds, especially those which are well-known.

COBOCTA 5 COLORED BY COMPONENT

The image above (made with Stella 4d, as are others in this post — software available here) is colored in the traditional style for compounds: each of the five cuboctahedra is assigned a color of its own. There’s a problem with this, however, and it is related to the triangular faces, due to the fact that these faces appear in coplanar pairs, each from a different component of the compound.

COBOCTA 5 COLORED TRIANGLE Face

The yellow regions above are from a triangular face of the yellow component, while the blue regions are from a blue triangular face. The equilateral triangle in the center, being part of both the yellow and blue components, must be assigned a “compromise color” — in this case, green. The necessity of such compromise-colors can make understanding the compound by examination of an image more difficult than it with with, say, the compound of five cubes (not shown, but you can see it here, if you wish). Therefore, I decided to look at this another way: coloring each face of the five-cuboctahedra compound by face type, instead of by component.

COBOCTA 5 COLORED BY FACE TYPE

Another helpful view may be created by simply hiding all the faces, revealing internal structure which was previously obscured.

COBOCTA 5 HOLLOW

Since the dual of the cuboctahedron is the rhombic dodecahedron, the dual of the compound above is the compound of five rhombic dodecahedra, shown, first, colored by giving each component a different color.

RD 5 colored by component

A problem with this view is that most of what’s “going on” (in the way the compound is assembled) cannot be seen — it’s hidden inside the figure. An option which helped above (with the five-cuboctahedra compound), coloring by face type, is not nearly as helpful here:

RD 5 colored by face type

Why wasn’t it helpful? Simple: all sixty faces are of the same type. It can be made more attractive by putting Stella 4d into “rainbow color” mode, but I cannot claim that helps with comprehension of the compound.

RD 5 colored rainbow

With this compound, what’s really needed is a “ball-and-stick” model, with the faces hidden to reveal the compound’s inner structure.

RD 5 colored hollow

Since the two five-part compounds above are duals, they can also be combined to form a ten-part compound: that of five cuboctahedra and five rhombic dodecahedra. In the first image below, each of the ten components is assigned its own color.

Compound of 5 Cuboctahedra and dual colored by component

In this ten-part compound, the coloring-problem caused in the first image in this post, coplanar and overlapping triangles of different colors, vanishes, for those regions of overlap are hidden in the ten-part compound’s interior. This is one reason why this coloring-scheme is the one I find the most helpful, for this ten-part compound (unlike the two five-part compounds above). However, so that readers may make this choice for themselves, two other versions are shown below, starting with coloring by face type.

Compound of 5 Cuboctahedra and dual colored by face typet

Finally, the hollow version of this ten-part compound. This is only a personal opinion, but I do not find this image quite as helpful as was the case with the five-part compounds described above.

Compound of 5 Cuboctahedra and dual colored rainbow

Which of these images do you find most illuminating? As always, comments are welcome.

Four-Part Compound of the Icosahedron, the Dodecahedron, the Cuboctahedron, and the Rhombic Dodecahedron

Compound of Cubocta and Dodeca and RD and Icosa

This compound was created using Stella 4d, software you can try for yourself here.

A Platonic/Catalan Compound and Its Dual, a Platonic/Archimedean Compound

Compound of Rhombic Dodeca and Icosa

Shown above: the compound of the icosahedron and the rhombic dodecahedron. Below is its dual, the compound of the dodecahedron and the cuboctahedron.

Compound of Dodeca and Cubocta

Both these compounds were created using the “add/blend polyhedron from memory” function in Stella 4d: Polyhedron Navigator. To check out this program for yourself, just follow this link.