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.
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 56 CRF polychora by all possible combinations of {tetrahedron, cube, octahedron, dodecahedron, 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 54 new CRF polychora. This brings the running total to 210.
The remaining CRF polyhedra cannot generate pyramidal forms for one (or both) of the following reasons:
- the polyhedron contains a contour with at least six edges, thus any pyramid of it would require base-apex edge lengths longer than base-base edge lengths, and thus not be CRF;
- 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.
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.
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).
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.
Augmented duoprisms
The duoprisms are a source of a potentially very large number of CRF polychora, 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=38. Many different subsets of pentagonal prisms can be augmented in this way while still remaining convex, leading to a very large number of combinations.
