# Pyrorectichoron (EntityTopic, 11)

### From Hi.gher. Space

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 • dicone • coninder |

Torii:
| tiger • torisphere • spheritorus • torinder • ditorus |