# 5-orthoplex

5-orthoplex | |
---|---|

Rank | 5 |

Type | Regular |

Notation | |

Bowers style acronym | Tac |

Coxeter diagram | o4o3o3o3x () |

Schläfli symbol | {3,3,3,4} |

Bracket notation | <IIIII> |

Elements | |

Tera | 32 pentachora |

Cells | 80 tetrahedra |

Faces | 80 triangles |

Edges | 40 |

Vertices | 10 |

Vertex figure | Hexadecachoron, edge length 1 |

Petrie polygons | 192 skew decagonal-decagrammic coils |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Face radius | |

Cell radius | |

Inradius | |

Hypervolume | |

Diteral angle | |

Height | |

Central density | 1 |

Number of external pieces | 32 |

Level of complexity | 1 |

Related polytopes | |

Army | Tac |

Regiment | Tac |

Dual | Penteract |

Conjugate | None |

Abstract & topological properties | |

Flag count | 3840 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | B_{5}, order 3840 |

Flag orbits | 1 |

Convex | Yes |

Net count | 9694 |

Nature | Tame |

The **5-orthoplex**, also called the **pentacross**, **triacontaditeron**, **tac**, or **square-octahedral duotegum**, is a regular 5-polytope. It has 32 regular pentachora as facets, joining 16 to a vertex in a hexadecachoral arrangement. It is the 5-dimensional orthoplex.

It can also be seen as a convex segmentoteron, as a pentachoric antiprism. It is also the hexadecachoric tegum.

## Vertex coordinates[edit | edit source]

The vertices of a regular 5-orthoplex of edge length 1, centered at the origin, are given by all permutations of:

- .

## Representations[edit | edit source]

A 5-orthoplex has the following Coxeter diagrams:

- o4o3o3o3x () (full symmetry)
- o3o3o3x *b3o () (D
_{5}symmetry, has demitesseract verf) - xo3oo3oo3ox&#x (A
_{4}axial, pentachoric antiprism) - ooo4ooo3ooo3oxo&#xt (B
_{4}axial, as hexadecachoric bipyramid) - qo oo4oo3oo3ox&#zx (B
_{4}×A_{1}symmetry, hexadecachoric bipyramid) - oxo3ooo3oo *b3oo&#zx (D
_{4}symmetry, demitesseractic bipyramid) - qo ox3oo3oo *c3oo&#zx (D
_{4}×A_{1}symmetry, demitesseractic bipyramid) - xox ooo4ooo3oxo&#xt (B
_{3}×A_{1}symmetry, edge-first) - xox ooo3oxo3ooo&#xt (A
_{3}×A_{1}axial, edge-first) - xoo3oox oxo4ooo&#xt (B
_{2}×A_{2}symmetry, face-first) - oxo oxo xoo3oox&#xt (B
_{2}×K_{2}axial, triangle-first) - xoo3ooo3oox oqo&#xt (A
_{3}×A_{1}symmetry, cell-first) - oxoo3oooo3ooox&#xr (A
_{3}symmetry) - xoxo oxoo3ooox&#xr (A
_{2}×A_{1}symmetry) - xo4oo oo4oo3ox&#zx (B
_{3}×B_{2}symmetry, square-octahedral duotegum) - xo xo oo3ox3oo&#zx (A
_{3}×K_{2}symmetry, square-octahedral duotegum) - o(xo)o o(xo)o o(ox)o o(ox)o&#xt (B
_{2}×B_{2}symmetry, square duotegmatic bipyramid)

## Related polytopes[edit | edit source]

The regiment of the 5-orthoplex contains 4 uniform members, including itself, one with D5 symmetry (the hexadecahemidecateron), and 2 with pentachoric antiprism symmetry (the pentachoric hemiantiprism and spinopentachoric hemiantiprism). There are also 2 scaliform members known with 5-2 step prism alterprismatic symmetry.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Primary Polytera" (#3).

- Klitzing, Richard. "tac".
- Wikipedia contributors. "5-orthoplex".
- Hi.gher.Space Wiki Contributors. "Aeroteron".