Some Variants of Kepler’s Stella Octangula

The Stella Octangula is also known as the compound of two tetrahedra, which works well because the tetrahedron is self-dual. All of these are also two-part compounds, with varying amounts of similarity to the Stella Octangula. The first one is also the 26th stellation of the triakis octahedron, one of the Catalan solids.

compound and 26th stellation of triakis octahedron

All of these were made using Stella 4d, which may be tried or purchased at http://www.software3d.com/Stella.php.

odd compound

SO var d

SO var sdd

SO variant

Two Compounds of Six Tetrahedra Each

compound of six elongated tetrahedra

In the image above, which I stumbled upon using Stella 4d (available here), the tetrahedra are elongated. If they are regular, instead, the same arrangement looks very different:

Tetrahedra 6

Two Colorings of a Hollow Stella Octangula

hollow stella octangula 2

hollow stella octangula

Both of these versions of the stella octangula, or compound of two tetrahedra, were made with Stella 4d, software available at http://www.software3d.com/Stella.php.

An Elongated Stella Octangula

Image

An Elongated Stella Octangula

The Stella Octangula is another name for the compound of two tetrahedra. I made this elongated version, which uses narrow isosceles triangles in place of the usual equilateral triangles, using Stella 4d — polyhedron-manipulation software you can find at http://www.software3d.com/Stella.php.

A Variant of Kepler’s Stella Octangula

Image

A Variant of Kepler's Stella Octangula

Johannes Kepler named the compound of two tetrahedra the “stella octangula,” thus helping make it one of the best-known polyhedral compounds today. This variant uses triakis tetrahedra in place of the Platonic tetrahedra in that compound. The triakis tetrahedron is a Catalan solid, and is dual to the truncated tetrahedron.

Software credit: see http://www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.

600 Undulating Tetrahedra

Image

600 Undulating Tetrahedra

This is a 600-cell, one of the regular polychora (four-dimensional polytopes), with its edges and vertices rendered invisible, and its cells shrunk so that they do not touch. It’s rotating in hyperspace, and what you are seeing at any given moment is a particular three-dimensional “shadow,” or projection, of the entire figure.

It’s easy to make this sort of thing with software called Stella 4d, written by an Australian friend of mine. Here’s a link to a site where you can try it, as a free trial download, before deciding whether or not to purchase the fully-functioning version: http://www.software3d.com/Stella.php.

The Hyperspace Analogue of the Stella Octangula

The simplest polyhedron is the tetrahedron, and it is self-dual. The compound of two tetrahedra puts these duals together, and is most often called the Stella Octangula, a name Johannes Kepler gave it in the early 17th Century.

Image

In hyperspace, or 4-space, the simplest polychoron is the pentachoron, or 5-cell. Like the tetrahedron in 3-space, it is also self-dual. Here is the compound of two of them: hyperspace’s version of the Stella Octangula.

Compound of 1-Pen, 5-cell, Pentachoron and dual

Website to find the software used to make these images:  www.software3d.com/stella.php