Hecatonicosoctaexon

Hecatonicosoctaexon
(OFF file)
Rank	7
Type	Regular
Space	Spherical
Bowers style acronym	Zee
Info
Coxeter diagram	o4o3o3o3o3o3x
Schläfli symbol	{3,3,3,3,3,4}
Symmetry	BC7, order 645120
Army	Zee
Regiment	Zee
Elements
Vertex figure	Hexacontatetrapeton, edge length 1
Exa	128 heptapeta
Peta	448 hexatera
Tera	672 pentachora
Cells	560 tetrahedra
Faces	280 triangles
Edges	84
Vertices	14
Measures (edge length 1)
Circumradius	√2/2 ≈ 0.70711
Inradius	√14/14 ≈ 0.26726
Hypervolume	√2/630 ≈ 0.0022448
Diexal angle	acos(–5/7) ≈ 135.58470°
Height	√14/7 ≈ 0.53452
Central density	1
Euler characteristic	2
Related polytopes
Dual	Hepteract
Conjugate	Hecatonicosoctaexon
Properties
Convex	Yes
Orientable	Yes
Nature	Tame

The hecatonicosoctaexon, or zee, also called the heptacross or 7-orthoplex, is one of the 3 regular polyexa. It has 128 regular heptapeta as facets, joining 4 to a pentachoron peak and 64 to a vertex in a hexacontatetrapetal arrangement. It is the 7-dimensional orthoplex. It is also a convex segmentoexon, as a heptapetal antiprism.

Vertex coordinates

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

• (±√2/2, 0, 0, 0, 0, 0, 0).

Representations

A hecatonicosoctaexon has the following Coxeter diagrams:

• o4o3o3o3o3o3x (full symmetry)
• o3o3o *b3o3o3o3x (D7 symmetry)
• xo3oo3oo3oo3oo3ox&#x (A6 axial, heptapetal antiprism)
• ooo4ooo3ooo3ooo3ooo3oxo&#xt (BC6 axial, hexacontatetrapetal bipyramid)
• qo oo4oo3oo3oo3oo3ox&#zx (BC6×A1 symmetry)
• xox ooo4ooo3ooo3ooo3oxo&#xt (BC5×A1 axial, dege-first)
• xox ooo3ooo3ooo *b3ooo3oxo&#xt (D5×A1 axial, edge-first with half symmetry)
• xoox oxoo3oooo3oooo3ooxo&#xr (A4×A1 symmetry)
• xoo3oox ooo4ooo3ooo3oox&#xt (BC4×A2 axial, triangle-first)
• xoo3oox oxo3ooo3ooo *d3ooo&#xt (D4×2 symmetry, triangle-first with half symmetry)
• oxo4oooo xoo3ooo3ooo3oox&#xt (A4×BC2 axial, pentachoron-first)
• oxo oxo xoo3ooo3ooo3oox&#xt (A4×A1×A1 symmetry, pentachoron-first with half symmetry)
• xo4oo oo4oo3oo3oo3ox&#zx (BC5×BC2 symmetry, square-triacontaditeral duotegum)
• xoo3ooo3oox ooo4ooo3oxo&#xt (BC3×A3 axial, tetrahedron-first)
• xoo3ooo3oox ooo3oxo3ooo&#xt (A3×A3 axial, tetrahedron-first with half symmetry)
• oo4oo3xo oo4oo3oo3ox&#zx (BC4×BC3 symmetry, octahedral-hexadecachoric duotegum)
• oo3xo3oo ox3oo3oo *e3oo&#zx (D4×A3 symmetry, tetratetrahedral-demitesseractic duotegum)
• oxoo3oooo3oooo3oooo3ooxo&#xr (A5 symmetry)
• oqo xoo3ooo3ooo3ooo3oox&#xt (A5×A1 axial, hexateron-first)
• xooo3ooxo oxoo3oooo3ooox&#xr (A3×A2 symmetry)
• ooxoo4ooooo oxooo3ooooo3oooxo&#xcr (A3×BC2 symmetry)
• oxooo3oooxo ooxoo3ooooo4ooooo&#xcr (BC3×A2 symmetry)
• o(xoo)o o(xoo)o o(oxo)o o(oxo)o o(oox)o o(oox)o&#xt (square triotegum bipyramid)

External links

• Klitzing, Richard. "Zee".

• Wikipedia Contributors. "7-orthoplex".