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.

The 92 non-uniform CRF polyhedra are known as the Johnson solids, and the list has been proven to be complete. Research to discover CRF polytopes in higher dimensions is ongoing; there are likely to be at least several billion[1][2] CRF polychora, if not more, many of which are diminishings of uniform polychora of the H4 family.

They encompass the set of Blind polytopes, which are the set of non-regular convex polytopes with regular facets. Those had been enlisted up to the various diminishings of the hexacosichoron already in the 1990s by the couple Gerd and Roswitha Blind. - Both, the CRFs and the Blind polytopes are direct extrapolations of the Johnson solids to higher dimensions. Needless to say that the CRFs define a much broader class.

  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.