CRF polychora discovery project (Meta, 13)

From Hi.gher. Space

See also: List of convex regular-faced polyhedra

This page documents an ongoing project to discover as many CRF polychora as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.

Other than infinite series, the non-Johnson CRF polyhedra are the regular polyhedra and the Archimedean polyhedra. The Johnson solids can be roughly divided into four categories: prismatoids (includes pyramids, cupolae, rotundae, and elongated/bi/gyro forms of the above), augmented polyhedra, diminished polyhedra and crown jewels. We will use the same categorizations here.

Richard Klitzing enumerated a list of 177 segmentochora, which are the orbiform CRF polychora. However, these overlap with several different categories below, so the segmentochora will not be considered a category of their own. TODO: find out how many segmentochora are not yet counted below.

Discovery index (D numbers)

As of February 2014, new CRF discoveries are assigned a discovery index (aka D number), as a way of uniquely identifying the discovery without committing to a specific categorization or naming of it, which may not be feasible due to insufficient information at the time of discovery.

The discovery index page serves as the authoritative list of D number assignments.


Total in this section (excluding stacks): 264


Main article: CRFP4DP/Diminishings
  • Some diminishings of uniform polychora produce various segmentotopes and CRF polychora.
  • There are many BT polychora.
  • There are three known non-uniform convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
  • There are 18 diminishings of the xylochoron (not 19, because one is the tesseract).
  • The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
  • The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
  • There are some modified bisected 600-cells, lunae (wedge-like multiply-bisected 600-cells) and rotundae.


Main article: CRFP4DP/Augmentations
  • The pyrochoron has a single augmentation, consisting of two pyrochora joined cell-to-cell.
  • The tesseract has 14 augmentations with cubical pyramids, one of which corresponds with the xylochoron. However, these augmentations are also a subset of the xylochoron's diminishings, so their count should not be included under the present category.
  • There are some modified tesseract augmentations and augmentations of truncated tesseracts.
  • There are probably many other augmentable uniform polychora, these have yet to be explored.
  • The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
  • Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress. Preliminary calculations indicate that augmentation with n-gonal magnabicupolic rings number in the millions, due to combinatorial explosion.


Modified Stott expansions

Arguably, the CRFs described below could be classified as crown jewels, even though since their initial discovery they have been found in large numbers, which somewhat defeats the label "crown jewels".

Partial Stott-expansions

Main article: Partial Stott-expansion

Klitzing discovered in 2013 that some infinite families can be expanded according to a lower symmetry group, giving new polytopes. Expanding a 16-cell by a 4-fold subsymmetry, for example, produces a CRF polychoron best described as the convex hull of a tesseract and an octagon, both centered on the origin. This represents the first step in a series of partial Stott expansions that eventually yields the runcinated tesseract (x4o3o3x).

EKF polytopes

In 2014 quickfur discovered a derivation of a bilbiro from an icosahedron. This led to the discovery of various partial expansions of the hydrochoron. This derivation is not merely a partial Stott expansion, but a modification of partial Stott expansion by adding an initial faceting step before the actual Stott expansion. Student91 discovered an underlying general scheme where a polytope (usually uniform or regular) can be partially faceted according to some subsymmetry, and then Stott-expanded according to the same symmetry in order to restore convexity, often resulting in novel CRFs. The faceting is done by "punning" a CD node label with an equivalent label that has a negative value, thereby producing a non-convex faceting, which is then restored to 0 by the subsequent Stott expansion.

In 3D, the icosahedron, for example, can be faceted according to a 2-fold subsymmetry and then expanded, producing a bilbiro (J91). The same process applied to a 3-fold subsymmetry produces the triangular hebesphenorotunda (J92). Applied to a 5-fold subsymmetry, this process produces J32, the pentagonal orthocupolarotunda.

This process has come to be known by the acronym EKF (expando-kaleido-faceting). When applied to various 4D uniform polytopes, primarily those in the 120-cell family, the EKF operation produces a large number of new CRFs, many of them containing J32, J91, and/or J92 cells. A good number, probably the majority, of crown jewels in the D4.x numbering scheme are EKF polytopes.

Infinite families

Main article: CRFP4DP/Infinite families
  • The obvious infinite family is that of the m,n-duoprisms (mn ≥ 3).
  • There is also an infinite family of prisms of the n-gonal antiprisms.
  • Mrrl discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two square antiprisms, one cube, four tetrahedra and four square pyramids. Details can be found in this post. In general, members of this family consists of two n-gonal antiprisms and an n-gonal prism, forming a 3-membered ring, with n tetrahedra and n square pyramids filling in the lateral gaps, for all n ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of biantiprismatic rings, and the specific term is the n-gonal biantiprismatic ring, e.g. square biantiprismatic ring. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the wedges.
  • The n-gonal pyramid antiprisms (n-gonal pyramid || inverted gyro n-gonal pyramid) are CRF, and for n=4 and n=5, non-orbiform. (For n=3, it is identical to the 16-cell.) They are identical to the n-antiprism bipyramid.

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