Bilbirothawroid (EntityClass, 12)

From Hi.gher. Space

(Difference between revisions)
(triacontasphenorotunda and diminished form should get separate numbers, add links, use elemental names, other minor edits)
m (→4D: actually diminished form should be first)
Line 21: Line 21:
* ['''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.
* ['''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.
* ['''CJ4.4'''] 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'''] <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.
+
* ['''CJ4.5.2'''] <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 ['''CJ4.5.1'''] 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.
:* ['''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.
:* ['''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.
:* ['''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).
:* ['''CJ4.5.4'''] 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&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.
* ['''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 18:17, 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.
  • [CJ4.5.2] (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 [CJ4.5.1] 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.
Retrieved from "http://hi.gher.space/wiki/Bilbirothawroid"

Pages in this category (14)