CRFP4DP/More prismatoids (Meta, 14)

From Hi.gher. Space


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.

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.


A further 12 CRF polychora can be generated by all possible combinations of {J32, J91, J92} × {pseudopyramid, psuedobipyramid, elongated pseudopyramid, elongated pseudobipyramid}. Here, a pseudopyramid means the apex is a pentagon in the case of J32, a digon in the case of J91, and a triangle in the case of J92. The first case (J91 || digon) was discovered on February 23, 2014 by student91, and the remaining forms were discovered soon after by Klitzing and Marek14. The J32 pseudopyramid was constructed by Klitzing by on March 31, 2014 based on an idea by quickfur. They have been verified to be CRF by quickfur.

All three pseudopyramids (J32, J91, J92) have height E/(2*phi) where E is the edge length, and arise from the Stott expansion of a suitably faceted icosahedral pyramid.