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.
Some examples of CRF polychora include:
The bilunabirotunda pseudopyramid, a cute little CRF polychoron derived from the icosahedral pyramid.
The tetrahedral ursachoron, a generalization of the tridiminished icosahedron to 4D.
The swirlprismatodiminished rectified 600-cell, a remarkable CRF polychoron whose surface consists of 12 rings of alternating pentagonal prisms and antiprisms and 20 helical rings of square pyramids, in swirlprism symmetry.
The study of 4D polytopes (polychora) goes back to 1852, when Ludwig Schläfli enumerated the six 4D convex regular polytopes (regular polychora), along with some of the non-convex regular polychora. Edmund Hess in 1883 completed the list of non-convex regular polychora. These regular polytopes represent the most restrictive class of 4D polytopes, in which all surface elements are transitive, that is, equivalent under their symmetry group.
A slightly larger class of polytopes was studied by Thorold Gosset in 1900: the semiregular polychora, which had regular polyhedra as cells, but with the relaxed requirement that the polychoron was not required to be regular, but only vertex-transitive. Alicia Boole Stott expanded the field by studying a wider class of polychora, the uniform polychora, in which the requirement on cell shapes was relaxed to allow Archimedean solids or prisms, while still retaining vertex transitivity. This study was completed in 1965 by John Conway and Michael Guy, who enumerated all the convex uniform polychora via computer search. Their list was proved to be complete by Marco Möller in 2004.
Meanwhile, in 1980, G. Blind, et al, enumerated the complete list of convex polychora whose cells are regular (i. e., Platonic solids), with no other requirements imposed. A large number of these are the non-adjacent diminishings of the 600-cell, which produce polychora with icosahedra and tetrahedra as cells. M. D. Sikirić and W. Myrvold determined in 2007 that there are 314,248,344 such diminishings, up to isomorphism.
Further generalizations proceeded in two other directions. In 2001, Dr. Richard Klitzing published the list of convex segmentochora, dropping the requirements of vertex transitivity and regular cell shapes, but restricting the scope of his study to the polychora whose vertices lie on two parallel hyperplanes and could be inscribed in a 3-sphere. These segmentochora represent the simplest non-uniform CRF polychora. A slightly different direction was taken in 2005, where the requirement of vertex transitivity was kept, but cells were permitted to be any shape as long as all edges are unit length. These form the class of scaliform polychora.
In 2009, a thread on Johnsonian polytopes was started on the Tetraspace forum, where “Johnsonian” was defined to be any polytope that (1) is convex, (2) is not vertex-transitive, and (3) has 2D surface elements that are all regular polygons. Later on, requirement (2) was dropped, producing the present definition of CRF polytopes. This class represents the most permissive generalization of the Johnson solids to higher dimensions, encompassing the convex regular and uniform polytopes, the Blind polytopes, Klitzing's segmentochora, convex members of the scaliform polychora, and a vast array of other shapes. The CRF polychoron discovery project was founded to find all 4D polytopes that fit under this definition, with the long-term goal of proving that the enumeration is complete.
Initially, the CRF polychora found by the project were simple combinations or modifications of previously-known polychora, such as convex stacks of Klitzing's segmentochora, or various augmentations and diminishings of the uniform polychora, and the “cut-and-paste” combinations thereof. These CRFs exhibited relatively simple Johnson solids as cells: pyramids, cupolae, rotundae, along with some simple uniform polyhedra such as prisms and antiprisms. It was conjectured that the more unusual Johnson solids, especially those toward the end of Norman Johnson's list that cannot be derived from the uniform polyhedra by simple “cut-and-paste” operations, would prove to be difficult to build a CRF polychoron out of, because their unusual shapes would make it difficult to close up the polytope in a way that met the CRF requirements, except in the trivial construction of their prisms.
On February 4, 2014, however, a CRF polychoron with icosahedral symmetry was discovered featuring bilunabirotundae (J91) as cells. Dubbed the castellated rhombicosidodecahedral prism, it was the first non-trivial example of a CRF that contained J91 cells. Spurred by this discovery, another CRF polychoron was discovered two days later, this time sporting four triangular hebesphenorotundae (J92) as cells. It was dubbed the triangular hebesphenorotundaeic rhombochoron. These two discoveries subsequently led to the discovery of a large class of CRF polychora having either J91 and J92 as cells, sometimes both. As it turns out, both J91 and J92 have deep connections to the icosahedron that lead to many CRF constructions derived from the uniform polychora in the 120-cell/600-cell family.
As of this writing (March 2018), no CRF polychoron is known that contains as cells the Johnson solids with no obvious relation to the uniform polyhedra, such as the snub disphenoid (J84) or the sphenocorona (J86), besides their trivial 4D prisms. It is presently unknown whether such CRF polychora exist. Research is ongoing.
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 solids, 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
There are some CRF polychora that, analogous to the special Johnson solids at the end of Johnson's list, cannot be derived by some combination of cut-and-paste operations of uniform polychora and/or Klitzing's segmentotopes. These unusual CRF polychora are nicknamed crown jewels for their rarity and unusual beauty. So far, a handful of 4D crown jewels have been discovered, but research is still ongoing to find more of them, with the long-term goal of enumerating the complete list.
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.
In 2007, Sikirić and Myrvold showed that there are 314,248,344 non-adjacent diminishings of the 600-cell, up to isomorphism. This number does not include many CRFs produced by adjacent diminishings; so the total number of CRF 600-cell diminishings must be much higher. On top of this, many of the uniform polychora in the 600-cell family admit analogous diminishings as well, so the total number of CRFs in this category may well be several times that of the 600-cell.
The m,n-duoprisms can be augmented with n-gonal and m-gonal prism pyramids as well as 2n-prism||n-gon segmentochora, to form CRF polychora. The number of augmentations with prism pyramids alone is 1633 for various values of m and n up to the 5,20-duoprism. If the 2n-prism||n-gon segmentochora augmentations are included, preliminary research indicates a combinatorial explosion of close to 12 million CRF augmentations (up to isomorphism).