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 | B5, order 3840 |
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 () (D5 symmetry, has demitesseract verf)
- xo3oo3oo3ox&#x (A4 axial, pentachoric antiprism)
- ooo4ooo3ooo3oxo&#xt (B4 axial, as hexadecachoric bipyramid)
- qo oo4oo3oo3ox&#zx (B4×A1 symmetry, hexadecachoric bipyramid)
- oxo3ooo3oo *b3oo&#zx (D4 symmetry, demitesseractic bipyramid)
- qo ox3oo3oo *c3oo&#zx (D4×A1 symmetry, demitesseractic bipyramid)
- xox ooo4ooo3oxo&#xt (B3×A1 symmetry, edge-first)
- xox ooo3oxo3ooo&#xt (A3×A1 axial, edge-first)
- xoo3oox oxo4ooo&#xt (B2×A2 symmetry, face-first)
- oxo oxo xoo3oox&#xt (B2×K2 axial, triangle-first)
- xoo3ooo3oox oqo&#xt (A3×A1 symmetry, cell-first)
- oxoo3oooo3ooox&#xr (A3 symmetry)
- xoxo oxoo3ooox&#xr (A2×A1 symmetry)
- xo4oo oo4oo3ox&#zx (B3×B2 symmetry, square-octahedral duotegum)
- xo xo oo3ox3oo&#zx (A3×K2 symmetry, square-octahedral duotegum)
- o(xo)o o(xo)o o(ox)o o(ox)o&#xt (B2×B2 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".