Hecatonicosoctaexon
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Hecatonicosoctaexon | |
---|---|
![]() | |
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[edit | edit source]
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[edit | edit source]
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r (A3×BC2 symmetry)
- oxooo3oooxo ooxoo3ooooo4ooooor (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[edit | edit source]
- Klitzing, Richard. "Zee".
- Wikipedia Contributors. "7-orthoplex".