Bilbirothawroid (EntityClass, 12)
From Hi.gher. Space
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|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. | |CJ4.4||[[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. | ||
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| - | |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 rotunda]]e, 1 rhodopantohedron, 1 [[truncated dodecahedron]], and 20 [[triangular hebesphenorotunda|J92]]'s, for a total of 166 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 rotunda]]e, 1 rhodopantohedron, 1 [[truncated dodecahedron]], and 20 [[triangular hebesphenorotunda|J92]]'s, for a total of 166 cells. |
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|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. | |CJ4.5.2||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. | ||
Revision as of 23:53, 22 February 2014
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 | expanded castellated rhodoperihedral 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 J91 cells derived from the hydrocantichoron o5x3o3x. This CRF was successfully constructed by Quickfur on February 20, 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. |
