Circumradii of stellated polyhedra

Question

I'm constructing models of various stellated polyhedra. I'd like all of them to be the same size, i.e., have the same size circumsphere. In order to do this, I'd like to know the ratios of their circumradii to their edge lengths. I calculated this ratio for the stella octangula and got $\sqrt{6}$/4. The other polyhedra that I'm interest in (small stellated dodecahedron, great dodecahedron, and great icosahedron) aren't as straightforward. Is there some systematic way to make these calculations?

Answer 1

My website https://bendwavy.org/klitzing/home.htm is mainly suited to find various informations on all sorts of polytopes of various dimensions. Besides a detailed elaboration of their elemental incidences it provides also several geometrical informations as well for each individual one. Further some pages about more general topics could be found.

Wrt. the asked for stellated polyhedra we get the following circumradius-to-edge size ratios:

• small stellated dodecahedron: $\sqrt{\frac{5-\sqrt{5}}{8}} \approx 0.587785$
• great dodecahedron: $\sqrt{\frac{5+\sqrt{5}}{8}} \approx 0.951057$
• great icosahedron: $\sqrt{\frac{5-\sqrt{5}}{8}} \approx 0.587785$

Note that the same values of the small stellated dodecahedron and the great icosahedron are due to the fact that the latter is just an edge-respecting faceting of the former. The same applies to the great dodecahedron, as it is nothing but an edge-respecting faceting of the (convex) icosahedron.

You also will find on the download page several excel spreadsheets aiming to provide the circumradius directly from the respective Coxeter-Dynkin diagram of the according polytope. And right next to the links the asked for details about the searched for and therein implemented formulae is provided as well.

Further interesting links to start with most probably could be

• https://en.wikipedia.org/wiki/Polyhedron
• https://polytope.miraheze.org/wiki/Polytope

--- rk