I'd like to know a name/source for the following concept:

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. Let $G$ be its symmetry group, and let $F$ be the collection of (top-dimensional) faces of $P$. Note that $G$ acts on $F$. Let's agree to call $P$ a transitive polyhedron if the action of $G$ on $F$ is transitive.

Have these been studied? If so, what are they actually called? What other nice properties do they enjoy?

Here are some examples of transitive polyhedra:

Many thanks!


As mentioned (and linked) in the wikipedia page for Bipyramid and as mentioned in the properties box on the wikipedia for Trapezohedron, one term for the property is simply "face-transitive". When you click the link on the page for the Bipyramid, you learn another term for the property: "isohedral", and a term for the polyhedra with this property: "isohedra".

There are a lot of things to say about isohedra, one of the more famous ones being to what extent they could be considered the "fair dice". (See, for example, this MO question.)

  • $\begingroup$ Thank you! And you're right; I should have looked a bit more carefully. $\endgroup$ – neilme Dec 15 '13 at 0:15

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