Bilbirothawroid (EntityClass, 12)

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(Difference between revisions)
(→4D: student91's new CRF)
(triacontasphenorotunda and diminished form should get separate numbers, add links, use elemental names, other minor edits)
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* ['''CJ4.2.x'''] The non-icosahedral [[ursatope|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]]).
* ['''CJ4.2.x'''] The non-icosahedral [[ursatope|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 pyramid]]s. It also has a [[Stott-expanded]] variant ['''CJ4.3.2'''], first suggested by [[student91]], consisting of two parallel [[rhodopantohedra]], 30 bilunabirotundae, 40 [[triangular prism]]s, 24 [[pentagonal cupola]]e, and 72 [[pentagonal prism]]s.
* ['''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 pyramid]]s. It also has a [[Stott-expanded]] variant ['''CJ4.3.2'''], first suggested by [[student91]], consisting of two parallel [[rhodopantohedra]], 30 bilunabirotundae, 40 [[triangular prism]]s, 24 [[pentagonal cupola]]e, and 72 [[pentagonal prism]]s.
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* ['''CJ4.4'''] 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.
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* ['''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.
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* ['''CJ4.5.1'''] <b>(Tentative name)</b> 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 [[triangular hebesphenorotunda|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.
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* ['''CJ4.5.1'''] <b>(Tentative name)</b> 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 rotunda]]e, 1 rhodopantohedron, 1 [[truncated dodecahedron]], and 20 [[triangular hebesphenorotunda|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.
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:* ['''CJ4.5.2'''] 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.
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:* ['''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 prism]]s, 24 pentagonal rotundae, 180 square pyramids, 220 tetrahedra, 30 [[rhodomesohedra]], and two rhodopantohedra.
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:* ['''CJ4.5.3'''] 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).
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:* ['''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).
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* ['''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&nbsp;2, whose first coordinate is less than the Golden Ratio &phi; or is equal to 2, then push the remaining vertices together (i.e., if the first coordinate is ≥&phi; then subtract &phi;; if it's ≤-&phi; then add &phi;). The vertices whose original first coordinates were &phi; become coincident. Then new points x5o3A are added with first coordinates ±1, where A=6*&phi;. 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.
+
* ['''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&nbsp;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.

Revision as of 17:53, 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:

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.
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