A Cluster of Nine Octahedra, and Related Polyhedra

If one starts with a central octahedron, then augments each of its eight triangular faces with identical octahedra, this is the result.

9 Octahedra

It is then possible to augment each visible triangle of this cluster with yet more octahedra, which produces this result, in which some octahedra overlap each other.

Meta-9 Octahedra

After making this, I wanted to see its convex hull:  the smallest, tightest-fitting convex polyhedron which can contain a given non-convex polyhedron. (I use Stella 4d: Polyhedron Navigator to perform these manipulations of polyhedra, and this program makes this a fast and easy process. If you’d like to try this software, even as a free trial download, the website to visit is http://www.software3d.com/Stella.php.) Here’s what this convex hull, which bears a resemblance to the rhombcuboctahedron, looks like.

Convex hull of meta-9-octahedron

Looking for previously-unseen, and interesting, polyhedra, I then starting stellating this convex hull. I did find something interesting — to me, anyway — after only two stellations.

2nd stellation of Convex hull of meta-9-octahedron

That concluded my latest polyhedral investigation, but I certainly don’t intend it to be my last.

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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