Pyrorectichoron (EntityTopic, 11)
From Hi.gher. Space
m (2-pyrotomochoron moved to 3-pyrotomochoron: oops) |
Catisfluffy (Talk | contribs) (Add FLD) |
||
| (15 intermediate revisions not shown) | |||
| Line 1: | Line 1: | ||
| + | <[#ontology [kind topic] [cats 4D Uniform Polytope] [alt [[wikipedia:Rectified_5-cell]]]]> | ||
{{STS Shape | {{STS Shape | ||
| + | | image=<[#embed [hash 8PDKJVVWYTWKJ9MY5T4TAHV0XM] [width 150]]><br/>Petrie polygon | ||
| dim=4 | | dim=4 | ||
| elements=10, 30, 30, 10 | | elements=10, 30, 30, 10 | ||
| genus=0 | | genus=0 | ||
| ssc2=Kp2 | | ssc2=Kp2 | ||
| - | | extra={{STS Uniform polytope | + | | extra={{STS Polytope |
| + | | symbol=o3x3o3o | ||
| + | | flayout={{FLD|dim=4|left=e3|erev2|a3|line|a3|line2|a3|end}} | ||
| + | | dual=''Self-dual'' | ||
| + | | bowers=Rap | ||
| + | }}{{STS Uniform polytope | ||
| schlaefli=r{[[Triangle|3]],[[Tetrahedron|3]],[[Pentachoron|3]]}, t<sub>1</sub>{3,3,3} or t<sub>2</sub>{3,3,3} | | schlaefli=r{[[Triangle|3]],[[Tetrahedron|3]],[[Pentachoron|3]]}, t<sub>1</sub>{3,3,3} or t<sub>2</sub>{3,3,3} | ||
| + | | dynkin=o3x3o3o | ||
| vfigure=[[Triangular prism]] | | vfigure=[[Triangular prism]] | ||
| vlayout=([[Triangle|3]]<sup>[[Tetrahedron|3]]</sup>)<sup>2</sup>⋅([[Triangle|3]]<sup>[[Octahedron|4]]</sup>)<sup>3</sup> | | vlayout=([[Triangle|3]]<sup>[[Tetrahedron|3]]</sup>)<sup>2</sup>⋅([[Triangle|3]]<sup>[[Octahedron|4]]</sup>)<sup>3</sup> | ||
| - | |||
| - | |||
| - | |||
}}}} | }}}} | ||
| + | The '''pyrorectichoron''' is a uniform polychoron in the pyrotope family. Its cells are 5 [[tetrahedra]] and 5 [[octahedra]]. | ||
| - | + | The pyrorectichoron can be constructed as tetrahedron || octahedron, meaning it's also a [[segmentochoron]], K4.5. Deleting any vertex from the pyrorectichoron forms the [[trigonal biantiprismatic ring]], or K4.6, which can be constructed as [[trigonal prism]] || gyrated [[triangle]]. Deleting another vertex - specifically, one of the three from the gyrated triangle - will form the [[digonal gyrobicupolic ring]], or K4.8. | |
| - | [[ | + | == Images == |
| + | The following [[Petrie polygon]] and [[stereographic projection]] each highlight the same arrangement of two neighboring (at a point) tetrahedral cells in red and orange and the octahedral cell between them in blue: | ||
| + | <blockquote><[#embed [hash ZKCW86KPBE29PQQXSTB9M6TBCQ]]> <[#embed [hash CQCST077FMNDPQMWMFG5WG08X1] [height 240]]></blockquote> | ||
| + | In the Petrie polygon, as there are a total of 5 tetrahedra and 5 octahedra, the others can be obtained by rotating the above highlights by {{over|τ|5}}. | ||
| + | |||
| + | <[#polytope [id 128]]> | ||
| + | |||
| + | {{Tetrashapes}} | ||
Latest revision as of 17:56, 19 September 2017
The pyrorectichoron is a uniform polychoron in the pyrotope family. Its cells are 5 tetrahedra and 5 octahedra.
The pyrorectichoron can be constructed as tetrahedron || octahedron, meaning it's also a segmentochoron, K4.5. Deleting any vertex from the pyrorectichoron forms the trigonal biantiprismatic ring, or K4.6, which can be constructed as trigonal prism || gyrated triangle. Deleting another vertex - specifically, one of the three from the gyrated triangle - will form the digonal gyrobicupolic ring, or K4.8.
Images
The following Petrie polygon and stereographic projection each highlight the same arrangement of two neighboring (at a point) tetrahedral cells in red and orange and the octahedral cell between them in blue:
In the Petrie polygon, as there are a total of 5 tetrahedra and 5 octahedra, the others can be obtained by rotating the above highlights by τ∕5.
Incidence matrix
Dual: (dual of pyrorectichoron)
| # | TXID | Va | Ea | 3a | 3b | C1a | C2a | Type | Name |
|---|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | ||||||
| 1 | Ea | 2 | = digon | ; | |||||
| 2 | 3a | 3 | 3 | = triangle | ; tet-oct | ||||
| 3 | 3b | 3 | 3 | = triangle | ; oct-oct | ||||
| 4 | C1a | 4 | 6 | 4 | 0 | = tetrahedron | ; | ||
| 5 | C2a | 6 | 12 | 4 | 4 | = octahedron | ; | ||
| 6 | H4.1a | 10 | 30 | 20 | 10 | 5 | 5 | = pyrorectichoron | ; |
Usage as facets
This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
