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Naming abstract regular polyhedra

An abstract regular polytope is an abstract polytope whose automorphism group is flag-transitive. Due to being abstract, abstract polytopes don't need to be realized in an some space to be considered regular. As an example, the Klein map is an abstract regular polyhedron with 24 heptagons and 56 vertices, however it does not have a regular representation in E³, and is therefore not one of the usually considered 48 regular polyhedra. All regular polytopes are also abstractly regular.

This page constructs a naming system for all abstract regular polyhedra.

Naming system

The name of an abstract regular polyhedron consists of two parts: the name of the surface the polyhedron lies on and how many faces the polyhedron has.

If the surface does not have a proper name, it attempts to name the surface based on some well-known map that lies on that surface. For orientable surfaces, the list goes:

• 0: spherical, can be omitted
• 1: Heawood (after the Heawood map)
• 2: Bolza (after the Bolza surface)
• 3: Klein (after the Klein quartic)
• 4: Bring (after Bring's surface)
• 5: Sherk (after Sherk's map)
• 6: echidnic (after the echidnohedron)
• 7: Macbeath (after the Macbeath surface)

If the surface still cannot be named through this method, a more objective approach is taken; if the surface is orientable, the name of the surface is "(Latin numeral prefix) + toral", if it is non-orientable then the name is "Latin numeral + projective", with the numeral corresponding to the surface's genus. If the genus is 1, the numeral prefix is ignored.

The latter part simply consists of the Greek numeral representing the number of faces + -hedron.

Failsafes

In case the genus is not enough to fully distinguish a polyhedron, there are a few failsafes that will very likely distinguish between the problematic polyhedra, in this order:

• In case the two polyhedra have different face types, the adjective of the face type is added to the beginning: Sherk's map and the dual Fricke-Klein map (both Sherk icositetrahedra) -> square Sherk icositetrahedron and hexagonal Sherk icositetrahedron
• In case the two polyhedra have different vertex figures, the Latin numeral + verticed is added to the beginning: dual Gordan map and the dual Petrial Gordan map (both quintiprojective dodecahedra) -> quadriverticed quintiprojective dodecahedron and sexaverticed quintiprojective dodecahedron
• In case the two polyhedra have different Petrie polygons, the Latin numeral + petrie is added to the beginning: {7,3}₁₂ and {7,3}₁₄ (both [long name]) -> duodecapetrie [long name] and quattuordecapetrie [long name]

For examples of this naming scheme, go to Lithic polyhedra.