- 1. the core (unstellated) triakis tetrahedron.
- 2. the first stellation
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
- 9.
- 10. (three disphenoids)
- 11.
- 12.
- 13.
- 14. (jumping jacks)
- 15.
- 16.
- 17. the final stellation.
- 18.
- 19.
- 20.
- 21.

Here are the 21 fully supported stellations of the triakis tetrahedron (the simplest Archimedean dual). Of these, seventeen have full tetrahedral symmetry (with six mirror planes) and four are chiral. I find most of these to be quite beautiful. Take a look and see! Each is a kind of nonconvex dodecahedron.

As with other stellated polyhedra, there is no essential order for listing them. They appear here generally from the core out, but in a no particular order. The figure at right shows the one numbered 14 below, in which the star-shaped face shape resembles someone jumping with their arms stretched out. The whole thing strikes me as twelve skinny interpenetrating acrobats. In the final stellation, you'll find the same acrobats, but they have gained weight. (I'll leave you to find your own Freudean associations in the others.)

The above each have six mirror planes. The following four are chiral:

The only published references to these solids which I know of is a paper
by Anthony Smith, listed in the references,
which illustrates six of the reflexible cases, and pp. 36-37 of Magnus
Wenninger's *Dual Models*, which illustrates the first and last stellations.