Source
The Klein quartic is a surface of genus 3, i.e., isomorphic to a three-holed torus. It can be thought of as a hyperbolic quotient, in which case it has two regular tilings: the quartic icositetrahedron {7,3}8 and the quartic pentecontahexahedron {3,7}8, being duals of each other. When interpreted as a quotient of a hyperboloid in Minkowski space, they become Platonic solids in a sense: they are both vertex-, edge- and face-transitive and convex, being identical to the regular tilings {7,3} and {3,7}. These two regular polyhedra can then be modified in some ways to make the uniform "Archimedean" polyhedra of this space, which in turn can be used to make non-uniform convex regular faced "Johnson/CRF" polyhedra. This page lists all known convex regular faced polyhedra in the Klein quartic.
To be clear, "convex" here means that the polyhedron, when taken the universal cover of its space, is convex. This excludes the many n≥7 polygonal pyramids, although by this definition something like the pentagonal pyramid wouldn't be convex in the real projective plane as it'd be equivalent to the compound of two pentagonal pyramids.
As for their completeness, none of these categories have been proven to be complete. Categories II is relatively simple: the same algorithm used to count the Blind diminishings of the hexacosichoron may be used.[1] For the other three though, proving their completeness is much more complicated, especially for category I.
In the list pages, official Bowers acronyms are
All images in the list pages were created using HyperRogue.
Categories
Category I: Uniforms. These are the nine uniform polyhedra of the Klein quartic.
Category II: Klein diminishings. These are the 18 non-uniform CRF diminishings of the quartic pentecontahexahedron (klein), analogous to the four diminishings of the icosahedron.
Category III: Rotundae. These are the three diminishings of kuq, somewhat analogous to the pentagonal rotunda, after which they are named. They all have central octagonal faces.
Category IV: Sirkuq diminishings and gyrations. These are the 158 diminishings and gyrations of sirkuq, analogous to the diminishings and gyrations of the small rhombicosidodecahedron, calculated from the diminishings of klein. Not available yet.