# A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $$d$$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $$d\ge 4$$ and the following properties?

• All facets are congruent,
• it has an insphere (a sphere to which each facet is tangent to), and
• it is not facet-transitive.

In 3-dimensional space there is an example with the "memorable" name Pseudo-deltoidal icositetrahedron, depicted below. I believe its the only such polyhedron. I am not aware of any higher dimensional examples. • This seems possibly related to a previous MO question about irregular, but fair dice. Is your 3D example the same as the example described in this answer? mathoverflow.net/questions/46684/… Jun 3 '20 at 17:54
• @YoavKallus Interesting link! Yes that's exactly the same polyhedron. Jun 3 '20 at 18:14

Next, there is a series of examples described and pictured in my old question Can the sphere be partitioned into small congruent cells? . Each of these examples is what you want in $$R^3$$. If you begin with any one such example and place it on a great 2-sphere of the 3-sphere in $$R^4$$ (say, the "equator"), then suspend it from the poles, you will get an example answering your question. The construction generalizes inductively to all higher dimensions.