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:
- [CJ4.1] 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.
- [CJ4.2.x] 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).
- [CJ4.3.1] 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 [CJ4.3.2], first suggested by student91, consisting of two parallel rhodopantohedra, 30 bilunabirotundae, 40 triangular prisms, 24 pentagonal cupolae, and 72 pentagonal prisms.
- [CJ4.4] The triangular hebesphenorotundaeic rhombochoron (or J92-rhombochoron), containing four triangular hebesphenorotundae (J92) cells at 60° and 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.
- [CJ4.5.1] (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 [CJ4.5.2] has 30 tetrahedra, 60 square pyramids, 30 triangular prisms, 12 pentagonal cupolae, 12 pentagonal rotundae, 1 rhodopantohedron, 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.
- [CJ4.5.3] 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 rhodomesohedra, and two rhodopantohedra.
- [CJ4.5.4] On the same day, Marek suggested a "squashed" version of the previous CRF, in which the 30 rhodomesohedra are replaced with bilunabirotundae (J91). This was confirmed to be CRF by Quickfur. It has 2 rhodopantohedra, 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).
- [CJ4.6] On 18 February 2014, student91 described a CRF containing J91 cells derived from the cantellated 600-cell o5x3o3x. This CRF was successfully constructed by Quickfur on 20 February 2014 and verified to be CRF. Its construction is as follows: delete the vertices from o5x3o3x of edge length 2, whose first coordinate is less than the Golden Ratio φ or is equal to 2, then push the remaining vertices together (i.e., if the first coordinate is ≥φ then subtract φ; if it's ≤-φ then add φ). The vertices whose original first coordinates were φ become coincident. Then new points x5o3A are added with first coordinates ±1, where A=6φ. These modifications of o5x3o3x cause a reduction of its equatorial cells into 30 bilunabirotunda (J91's) in icosahedral symmetry, with the 12 gaps between them filled by circles of alternating hexagonal prisms and truncated tetrahedra surrounding a pentagonal prism with pentagonal cupolae above and below it. Twelve of the o5x3o's near the equatorial layer are truncated into pentagonal rotundae, and a number of cuboctahedra become triangular cupolae.
