# Isotopic polytope

An **isotopic polytope** or **facet-transitive polytope** is a polytope whose facets are identical under its symmetry group. In other words, given any two facets, there is a symmetry of the polytope that transforms one into the other. In an isotopic polytope, all of its facets are tangent to a hypersphere. The dual of an isotopic polytope is an isogonal polytope, which are made out of one vertex type. All regular polytopes are isotopic. Branko Grunbaum called isotopic polyhedra **isohedra**, and isotopic polychora are sometimes called **isochora**.

If an isotopic polytope is also isogonal, it is called a noble polytope. Self-dual isotopic polytopes are also noble.

Convex isotopic polytopes are fair dice, but there are fair dice that are not isotopic.

## In 3 dimensions[edit | edit source]

As stated above, the regular polyhedra (the Platonic solids and Kepler-Poinsot polyhedra) are isotopic. The duals of the uniform polyhedra are also isotopic, since uniform polytopes are defined as being isogonal. The Catalan solids (duals of the Archimedean solids, a subset of uniform polyhedra) are notable examples of isotopic polyhedra. Isotopic polyhedra can be referred to as "isohedral" or "face-transitive."

Disdyakis dodecahedron | Pentagonal hexecontahedron | Pentagonal antitegum | Medial rhombic triacontahedron |
---|---|---|---|

Catalan solid, dual of the truncated cuboctahedron | Catalan solid, dual of the snub dodecahedron | Dual of the pentagonal antiprism | Dual of the dodecadodecahedron |

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