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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!

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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.)

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  • $\begingroup$ 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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