The CRF Polychora
The CRF polychora are a generalization of the Johnson solids to 4D. They are 4D polytopes belonging to the general class of CRF polytopes, which have the following characteristics:
They are convex;
Their 2D faces are regular polygons.
The acronym CRF stands for Convex Regular-Faced. In 4D, these two characteristics imply that the cells of a CRF polychoron must be either regular polyhedra, uniform polyhedra, or Johnson solids. As such, the CRF polychora include the six convex regular polychora and the convex uniform polychora.
As of this writing, the classification of all CRF polychora is still an ongoing research. Since all the 4D convex uniform polytopes have been discovered, as proven by Conway, the interest now lies in the non-uniform members of this class.
In 2001, Dr. Richard Klitzing enumerated the full list of convex segmentochora: 4D polytopes whose vertices lie on two parallel hyperplanes and which can be inscribed in a 4D hypersphere. These include the prisms of 3D uniform polyhedra and a number of Johnson polyhedra, pyramids of many of these polyhedra, and other less-obvious shapes, including one infinite family.
The segmentotopes are useful fundamental “building blocks” that can be assembled into larger CRF polychora. For example, pyramids or pyramid-like segmentotopes with the same shape on their bases can be assembled into bipyramids or elongated bipyramids (by inserting a prism between the bases).
Modified Uniform Polychora
Many CRF polychora can be derived from the uniform polychora by various “cut-and-paste” operations.
For example, the 24-cell can be cut into two halves which are themselves CRF polychora. In this case, they happen to be segmentotopes (4.29, or octahedron || cuboctahedron, in Klitzing's list). However, it is also possible to cut off one vertex from the 24-cell to produce a CRF polychoron with 18 octahedra, 6 square pyramids, and a cube, which is not in Klitzing's list. Thus, many new CRFs not numbered among the segmentotopes can be produced this way.
Many uniform polychora can also be augmented by gluing on segmentochora or other CRF polychora. For example, we can glue 24 icosahedral pyramids to the snub 24-cell to form the 600-cell. Attaching less than 24 icosahedral pyramids leads to various intermediate forms, all of which are CRF polychora.
A very large number of CRF polychora can be produced by these cut-and-paste operations.
Other CRF polychora
As of this writing, it is not known whether there are CRF polychora that cannot be derived by some combination of cut-and-paste operations of uniform polychora and/or Klitzing's segmentotopes. Research to discover such CRFs (or prove their non-existence) is still ongoing.
Number of CRF Polychora
It is known that there are infinite families of CRF polychora, among which are the infinite family of uniform duoprisms, uniform antiprism prisms, and one non-uniform infinite family among Klitzing's segmentotopes. It is currently not known whether there are any more infinite families of CRF polychora, nor how many CRF polychora there are outside of infinite families.
However, it is known that the number of the latter must be very large, because many CRF polychora can be derived by augmentations and diminishings of large uniform polychora such as the 600-cell. The sheer number of possible combinations of augmentations and diminishings means that the lower limit of this number must be at least in the thousands, if not tens of thousands.
The m,n-duoprisms can be augmented with n-gonal and m-gonal prism pyramids to form CRF polychora. There are a total of 1633 such augmentations of m,n-duoprisms for various values of m and n. A large number of these are augmented 5,n-duoprisms for n ranging up to 20 (possible due to the shallowness of the pentagonal prism pyramid). Duoprisms can also be augmented with other segmentotopes, so 1633 is only a lower limit on the total number of CRF augmented duoprisms.
Some examples of CRF polychora include:
- The cubical pyramid.
- The square magnabicupolic ring.
- The biparabigyrated cantellated tesseract, an interesting CRF polychoron with 8 J37 cells.
- The bi-icositetradiminished 600-cell, an interesting CRF polychoron bounded by 48 tridiminished icosahedra with swirlprism symmetry, being both vertex- and cell-transitive.