A Fractured Octahedron

Sometimes, when using Stella 4d (available here) to make various polyhedra, I lose track of how I got from wherever I started to the final step. That happened with this fractured version of an octahedron.

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.

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.

The Compound of the Octahedron and the Small Stellated Dodecahedron

compound of the small stellated dodecahedron and the octahedron

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.

Faceted Dodeca and compound of ten elongated octahedra.gif

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

Faceted Dodeca and compound of ten elongated octahedra without edges and vertices.gif

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.

Faceted Dodeca one part of ten with edges and vertices.gif

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

Faceted Dodeca one part of ten.gif

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.

compound of ten regular octahedra.gif

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.

compound of ten cubes.gif

A Compound of an Octahedron and a Pyritohedral Dodecahedron

compound of a pyritohedral dodecahedron and an octahedron

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