Convex regular-faced polytope

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A CRF polytope, short for convex regular-faced polytope, is any strictly convex polytope whose faces are all regular. They are a superset of the Blind polytopes, which have the additional constraint of having all regular facets (cells must be Platonic solids in 4D, etc.).

Polyhedra[edit | edit source]

The CRF polyhedra comprise the convex uniform polyhedra and the 92 Johnson solids, a set proven complete.

Polychora[edit | edit source]

The nonuniform CRF polychora are a very large set. The non-uniform 4D Blind polytopes have been completely enumerated to be over 300 million, the majority of them being the special cuts of the 600-cell. The non-Blind CRF diminishings of the 600-cell are even more numerous, likely to number several billion.[1][2] Many more CRF polychora arise from cut-and-paste operations of the 4D uniforms.

Three infinite families are known of CRF polychora: the prism products of two regular polygons, the prisms of the convex uniform antiprisms, and the antifastegiums. It is not yet known if there are more such infinite families.

Crown jewels[edit | edit source]

The end of the list of Johnson solids comprises the "elementary Johnson solids," loosely defined as those having no obvious construction from convex uniform polyhedra nor operations such as pyramid and prism products. There is particular interest in discovering CRF polytopes that subjectively qualify as "elementary;" these are known as the crown jewels, a term with no formal definition.

In 4D, some segmentotopes fit this description. Some polychora are labeled as crown jewels due to containing cells not often found in CRF polychora, such as bilunabirotundae or triangular hebesphenorotundae. The ursatopes, which generalize the tridiminished icosahedron, have led to discoveries of some crown jewels.

References[edit | edit source]

  1. In 2008 Mathieu Dutour Sikirić and Wendy Myrvold finally managed to provide the number of polytopes in the last class of Blind polytopes, the diminishings of the hexacosichoron with regular facets only, to be 314,248,344. Furthermore there will be mutually intersecting diminishings, yielding several diminished icosahedron facets. And there are deeper diminishings as well, resulting in additional dodecahedra and pentagonal pyramids, or even icosidodecahedra. And finally one could apply several of those things to the Wythoffian derivatives of the hexacosichoron as well.
  2. Sikirić, Mathieu Dutour; Myrvold, Wendy (2008). "The Special Cuts of the 600-cell". Contributions to Algebra and Geometry. 49 (1): 269–275.