CRF polychora discovery project (Meta, 13)
From Hi.gher. Space
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.
Convex uniform polychora
The first 64 CRF polychora are the convex uniform polychora, which can be divided up into:
- 9 pyromorphs,
- 9 xylomorphs,
- 12 stauromorphs (not 15, because three were already covered as xylomorphs),
- 15 rhodomorphs,
- 17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph),
- the snub demitesseract and the grand antiprism.
Richard Klitzing's segmentotopes
Dr. Richard Klitzing enumerated the full set of segmentotopes: CRF polychora constructed by the convex hull of two lower-dimensional polytopes placed in parallel hyperplanes spaced appropriately so that the result will have equal-length edges.
Prisms of Johnson solids
There are 92 Johnson solids. Each one has a prism which is a CRF polychoron, bringing the running total to 156.
Prismatoid forms
We can generate 52 CRF polychora by all possible combinations of {tetrahedron, cube, octahedron, icosahedron, square antiprism, pentagonal antiprism, triangular prism, pentagonal prism, square pyramid, pentagonal pyramid, diminished icosahedron, metabidiminished icosahedron, tridiminished icosahedron} × {pyramid, bipyramid, elongated pyramid, elongated bipyramid}. However, two of these - the "tetrahedral pyramid" and the "octahedral bipyramid" - are already covered as the pyrochoron and the aerochoron respectively, leaving us with 50 new CRF polychora. This brings the running total to 206.
The remaining CRF polyhedra cannot generate pyramidal forms for one (or both) of the following reasons:
- the polyhedron's vertices are further from its center than its edge length, thus any pyramid of it would require base-apex edge lengths longer than base-base edge lengths, and thus not be CRF;
- note that this reason is implied if the polyhedron contains a contour with at least six edges, but the converse is not always true, e.g. in the case of the dodecahedron
- the polyhedron cannot be inscribed in a sphere, thus there is no point equidistant from all base points, thus any pyramid of it would have at least two different base-apex edge lengths, and thus not be CRF.
wintersolstice originally proposed a list containing more polyhedra than those listed above, but this was incorrect due to the above reasons. He acknowledged that there was a mistake with the list some time ago, most likely realizing the same argument that has been written above, but did not give this explanation at the time.
Cupolae of regular polyhedra
We can generate 21 CRF polychora from the possible combinations of {tetrahedral, cubic, dodecahedral} × {cupola, orthobicupola, gyrobicupola, elongated cupola, elongated orthobicupola, elongated gyrobicupola, antiprism}. There are an additional 8 forms constructed as {octahedral, icosahedral} × {cupola, orthobicupola, elongated cupola, elongated orthobicupola}, as these forms do not use both duals. This gives 29 shapes in total.
The elongated cubic orthobicupola is the same as the runcinated tesseract, leaving us with 28 new CRF polychora.
Each cupola is constructed as the spline from the base polyhedron to its extratruncate. In the case of gyrobicupolae, the "other end" of the polychoron is the dual of the base shape. In the case of antiprisms, the spline is directly from the base shape to its dual.
The ability to construct these shapes with regular faces needs to be checked.
Two copies of the icosahedron-dodecahedron antiprism can be fitted together by their dodecahedral bases; the relative sizes of the icosahedron and dodecahedron of equal edge length ensures that the result is convex, and therefore CRF. It consists of 2 icosahedral cells, 100 tetrahedra, and 24 pentagonal pyramids. An elongated form is obtained by inserting a dodecahedral prism. Both forms have augmented and biaugmented variants. The other polyhedron-dual antiprisms do not produce convex CRFs this way, so this is a unique combination giving 6 new CRF polychora in total.
Bicupolic rings
Nine CRF polychora are available from the possible combinations of {triangle, square, pentagon} × {ortho, gyro, magna}. Keiji discovered the ortho- and gyro- forms, and quickfur discovered the magna- form. Keiji has dubbed these shapes bicupolic rings in general, and the specific naming pattern is n-gonal formbicupolic ring, e.g. square orthobicupolic ring.
The ortho- and gyro- forms are constructed as in this post. The magna- forms are constructed as in this post (second-to-last paragraph).
Augmented duoprisms
The duoprisms are a source of 1633 CRF polychora via augmentation (erecting pyramids on their cells), 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.
The following lists the number of CRF polychora generated by augmenting duoprisms:
| 3,3-duoprism: | 3 |
|---|---|
| 3,4-duoprism: | 5 |
| 3,5-duoprism: | 11 |
| 3,6-duoprism: | 4 |
| 4,5-duoprism: | 17 |
| 4,6-duoprism: | 4 |
| 4,7-duoprism: | 4 |
| 4,8-duoprism: | 7 |
| 5,5-duoprism: | 35 |
| 5,6-duoprism: | 12 |
| 5,7-duoprism: | 17 |
| 5,8-duoprism: | 29 |
| 5,9-duoprism: | 45 |
| 5,10-duoprism: | 77 |
| 5,11-duoprism: | 15 |
| 5,12-duoprism: | 25 |
| 5,13-duoprism: | 30 |
| 5,14-duoprism: | 48 |
| 5,15-duoprism: | 63 |
| 5,16-duoprism: | 98 |
| 5,17-duoprism: | 132 |
| 5,18-duoprism: | 208 |
| 5,19-duoprism: | 290 |
| 5,20-duoprism: | 454 |
| Total: | 1633 augmentations |
The 4,4-duoprism is omitted here, because it coincides with the tesseract, the augmentations of which are covered under another category.
The sharp drop in the number of augmentations between the 3,5-duoprism and the 3,6-duoprism, between the 4,5-duoprism and the 4,6-duoprism, and between the 5,5-duoprism and the 5,6-duoprism is because pyramids of hexagonal (or higher) prisms cannot be CRF, since equilateral triangles tile the hexagon and so no hexagonal (or higher) pyramid can be formed without breaking the regular-faced requirement. Thus, only one of the duoprism's two rings can be augmented.
The drop between the 5,10-duoprism and the 5,11-duoprism is caused by the fact that adjacent pentagonal prism pyramids erected on an n-membered duoprism ring are no longer convex after n=10, so from the 5,11-duoprism onwards only non-adjacent augmentations are permitted, thus reducing the number of possible combinations. Adjacent augments on the 5,10-duoprism have pentagonal pyramid cells that are coplanar, thus merging into a pentagonal bipyramid.
Augments of the 5,20-duoprism have pentagonal pyramids coplanar with the adjacent pentagonal prism, so they merge into elongated pentagonal pyramids. If the next prism in the ring is also augmented, then another pentagonal pyramid is added to the coplanar cell, turning it into an elongated pentagonal bipyramid.
No other duoprisms can be augmented with CRF pyramids and still remain convex.
Diminished polychora
Some regular polychora can be diminished to give CRF polychora. The 24-cell can be diminished into the tesseract by removing 8 square pyramids. Removing less than 8 pyramids in various configurations generates a number of distinct diminished 24-cells. It is also possible to remove square pyramids that do not correspond with facets of the tesseract, this generates a few more CRF polychora not included in the tesseract construction.
The diminished 16-cell coincides with the octahedral pyramid (see Prismatoid forms above).
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).
Rotundae
So far, three CRF rotundae have been discovered.
Mrrl discovered that a CRF polychoron can be cut from the rectified 120-cell when diminishing the latter. This polychoron consists of 1 icosidodecahedron, 12 pentagonal rotundae, and 40 tetrahedra. It can be considered the 4D analogue of the 3D pentagonal rotunda.
A similar CRF rotunda can be obtained from the cantellated 600-cell by a similar cutting, producing a polychoron with 1 icosidodecahedron, 12 pentagonal rotundae, 42 pentagonal prisms, 20 cuboctahedra, and 20 triangular cupolae.
Mrrl also found that the top of the second rotunda can be diminished, to obtain another CRF rotunda with 1 truncated icosahedron, 12 pentagonal rotunda, 30 pentagonal prisms, and 40 triangular cupolae.
These rotundae have birotunda forms as well as their corresponding elongates.
Infinite families
The obvious infinite family is that of the m,n-duoprisms (m ≥ n ≥ 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 (they are known as wedges in Klitzing's terminology).
