Isotopic polytope

From Polytope Wiki
Revision as of 15:48, 6 November 2023 by Plasmath (talk | contribs)
Jump to navigation Jump to search

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

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."

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