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.

# Tag Archives: octahedron

# 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.

# The Compound of the Octahedron and the Small Stellated Dodecahedron

I made this rotating virtual model using *Stella 4d: Polyhedron Navigator*, which you can try for yourself at http://www.software3d.com/Stella.php. This solid is different from most two-part polyhedral compounds because an unusually high fraction of one polyhedron, the yellow octahedron, is hidden inside the compound’s other component.

# Some Ten-Part Polyhedral Compounds

While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.

Here’s what this compound looks like with the edges and vertices hidden:

Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.

Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.

I made all these polyhedral transformations using *Stella 4d*, a program you can try for yourself at this website. *Stella *includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).

The next thing I wanted to try was to make the octahedra regular. *Stella* has a function for that, too, and here’s the result: a compound of ten regular octahedra.

My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.

# A Compound of an Octahedron and a Pyritohedral Dodecahedron

I made this using *Stella 4d: Polyhedron Navigator*. You can try this program for free at http://www.software3d.com/Stella.php.

# 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.

# Augmenting the Octahedron with Octahedra, Repeatedly.

This is an octahedron.

If you augment each face of an octahedron with more octahedra, you end up with this.

One can then augment each triangular face of this with yet more octahedra.

Here’s the next iteration:

This could, of course, go on forever, but one more step in the series is all you will see here. I don’t want to get caught in an infinite loop.

Performing various manipulations of polyhedra is easy with *Stella 4d: Polyhedron Navigator*, which I used to make all five of these rotating images. If you’d like to try this program for yourself, just check out http://www.software3d.com/Stella.php.