Isotopic polytope

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An isotopic polytope or facet-transitive polytope is a polytope whose facets are identical under its highest symmetry group. 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.

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

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."

Examples of isotopic ("isohedral") polyhedra
Disdyakis dodecahedron Pentagonal hexecontahedron Pentagonal trapezohedron 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
Disdyakis dodecahedron.png
Hexecontaedro pentagonal-iso.jpg
Pentagonal Trapezohedron without Compound of Six Pentagonal Trapezohedron of 5.svg
DU36 medial rhombic triacontahedron.png