A nice person called Payton Asch sent me an email with an observation about the Herschel enneahedron:

It looks like the underlying polytope for the enneahedron is a triangular bipyramid (two tetrahedra stacked on top of each other) or the dual polytope would be a triangular prism.

In the case of the triangular bipyramid you would truncate each of the vertices around the “equator” deep enough until the truncated areas meet at a vertex.

I immediately wanted to make this. I’m no good at cutting things, so I went to my 3D printer. Here’s what I came up with:

While making the 3D model, I noticed something else: if you just cut a little bit off the corners, instead of all the way to the middle, you get a different shape. It has 14 vertices (an even number), isn’t bitpartite, and is Hamiltonian.

I’ve made a little interactive tool to show the continuum from bipyramid to Herschel enneahedron:

Payton noticed that this intermediate shape is none other than the associahedron! How cool!