Bilbirothawroid (EntityClass, 12)
From Hi.gher. Space
Crown jewels are a catch-all term for unusual CRF polytopes with unique structures that cannot be obtained from the uniform polytopes or other simpler CRFs by simple "cut-and-paste" operations.
3D
They are the unusual Johnson solids at the end of Johnson's list:
- Snub disphenoid (J84),
- Snub square antiprism (J85),
- Sphenocorona (J86),
- Augmented sphenocorona (J87),
- Sphenomegacorona (J88),
- Hebesphenomegacorona (J89),
- Disphenocingulum (J90),
- Bilunabirotunda (J91),
- Triangular hebesphenorotunda (J92).
4D
So far, the following crown jewels have been discovered:
- snubdis antiprism (cube || icosahedron): a crown jewel discovered by Klitzing and included in his list of segmentochora. It is considered a crown jewel because it involves the non-obvious placement of two polyhedra of different symmetry groups in parallel hyperplanes.
- The non-icosahedral ursachora are arguably crown jewels as well, since they cannot be derived from the uniform polychora by simple "cut-and-paste" operations, and their construction was non-obvious until Wendy first constructed the icosahedral variant (which happens to be a diminishing of the hydrochoron).
- The castellated rhodoperihedral prism: an unusual prism-like CRF discovered by Quickfur on February 4, 2014, consisting of two parallel rhodoperihedra sandwiching 30 bilunabirotundae, 40 tetrahedra, and 24 pentagonal pyramids. It also has a Stott-expanded variant, first suggested by student91, consisting of two parallel rhodopantohedra, 30 bilunabirotundae, 40 triangular prisms, 24 pentagonal cupolae, and 72 pentagonal prisms.
- The triangular hebesphenorotundaeic rhombochoron (or J92-rhombochoron), containing four triangular hebesphenorotundae (J92) cells at a 60°/120° dichoral angles to each other, 6 metabidiminished icosahedra (J62), 6 triangular prisms, 24 pentagonal pyramids, 30 square pyramids, and 12 tetrahedra, discovered by Quickfur on February 6, 2014 based on a suggestion given by student91 after a previous failed attempt to construct a CRF with J92 cells.
- (Tentative name) The triacontasphenorotunda and its diminished form, first described by student91 on 13 February 2014, based on an idea by Quickfur that a CRF with J92 cells might exist as a modified diminishing of the rectified 120-cell. The diminished form has 30 tetrahedra, 60 square pyramids, 30 triangular prisms, 12 pentagonal cupolae, 12 pentagonal rotundae, 1 great rhombicosidodecahedron, 1 truncated dodecahedron, and 20 J92's, for a total of 166 cells. The non-diminished form has 70 tets + 20 J92's + 13 ids + 12 pecues + 60 squippies + 30 trips + 1 grid.
- On 15 February 2014, Quickfur constructed a related, as yet unnamed, CRF from the remainder of the rectified 120-cell after the above modified diminishing, which contains 40 J92's, 24 decagonal prisms, 24 pentagonal rotundae, 180 square pyramids, 220 tetrahedra, 30 icosidodecahedra, and two great rhombicosidodecahedra.
- On the same day, Marek suggested a "squashed" version of the previous CRF, in which the 30 icosidodecahedra are replaced with bilunabirotundae (J91). This was confirmed to be CRF by Quickfur. It has 2 great rhombicosidodecahedra, 40 J92's, 30 J91's, 36 decagonal prisms, and 240 square pyramids. It is the first known crown jewel that contains two 3D crown jewels as cells, the bilunabirotunda (J91), and the triangular hebesphenorotunda (J92).
