Isotopic polytope

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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. Branko Grunbaum called isotopic polyhedra isohedra, and isotopic polychora are rarely called isochora.

Notable subclasses of isotopic polytopes include the regular polytopes, Catalan solids (and uniform duals in general), trapezohedra, noble polytopes, isohedral deltahedra, gyrochora, and bipyramids of regular polygons. The dual of an isotopic polytope is an isogonal polytope, which are made out of one vertex type.

A finite, planar isotopic polytope has all of its facets tangent to an inscribed hypersphere.

Convex isotopic polytopes are fair dice.

In 3 dimensions[edit | edit source]

As stated above, the regular polyhedra (including 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".

Examples of isotopic ("isohedral") polyhedra
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