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:
| Index | Name | Discovered | Description |
|---|---|---|---|
| CJ4.1 | snubdis antiprism | 2001? by Klitzing | The cube || icosahedron was 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, and is the only crown jewel that is also a segmentotope. |
| CJ4.2.x | ursachora | ? by Wendy | Other than the icosahedral ursachoron, the ursatopes 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 | castellated rhodoperihedral prism | February 4, 2014 by Quickfur | An unusual prism-like CRF consisting of two parallel rhodoperihedra sandwiching 30 bilunabirotundae, 40 tetrahedra, and 24 pentagonal pyramids. |
| CJ4.3.2 | castellated rhodopantohedral prism | student91 | A Stott-expanded variant of CJ4.3.1 consisting of two parallel rhodopantohedra, 30 bilunabirotundae, 40 triangular prisms, 24 pentagonal cupolae, and 72 pentagonal prisms. |
| CJ4.4 | triangular hebesphenorotundaeic rhombochoron (or J92-rhombochoron) | February 6, 2014 by Quickfur | Contains 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. Based on a suggestion given by student91 after a previous failed attempt to construct a CRF with J92 cells. |
| CJ4.5.1 | diminished triacontasphenorotunda (tentative name) | February 13, 2014 by student91 | Based on an idea by Quickfur that a CRF with J92 cells might exist as a modified diminishing of the rectified 120-cell. 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. |
| CJ4.5.2 | triacontasphenorotunda (tentative name) | February 13, 2014 by student91 | Discovered at the same time as CJ4.5.1, the non-diminished form has 70 tets + 20 J92's + 13 ids + 12 pecues + 60 squippies + 30 trips + 1 grid. |
| CJ4.5.3 | none yet | February 15, 2014 by Quickfur | A related CRF constructed 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 | none yet | February 15, 2014 by Marek14 | 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 | none yet | February 18, 2014 by student91 | A CRF containing 30 J91 cells derived from the hydrocantichoron o5x3o3x. This CRF was successfully constructed by Quickfur on February 20, 2014 and verified to be CRF. |
| CJ4.7 | none yet | February 24, 2014 by student91 | An octa-diminishing of a modified rectified 600-cell (diminished) tristratic cap, containing 8 J92 lacing an icosidodecahedron and a truncated icosahedron on parallel hyperplanes. |
| CJ4.8 | none yet | February 25, 2014 by Quickfur | A CRF made from a tetrahedron-centered patch of the rectified 120-cell's surface, with modifications that create 4 J92 cells, 4 J91 cells, 4 tridiminished icosahedra (J63), and an assortment of other cells. |
| CJ4.9.x | none yet | February 22, 2014 by student91 | A family of related CRFs constructed from a compressed rectified 600-cell, wherein the vertex layers of the latter between two embedded truncated dodecahedra are deleted and the rest of the polytope compressed together, followed by the deletion of one or more decagon-decagon edges of the truncated dodecahedron to create J91 cells. |
