Hecatonicosoctaexon

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Hecatonicosoctaexon
7-orthoplex.svg
Rank7
TypeRegular
SpaceSpherical
Bowers style acronymZee
Info
Coxeter diagramo4o3o3o3o3o3x
Schläfli symbol{3,3,3,3,3,4}
SymmetryBC7, order 645120
ArmyZee
RegimentZee
Elements
Vertex figureHexacontatetrapeton, edge length 1
Exa128 heptapeta
Peta448 hexatera
Tera672 pentachora
Cells560 tetrahedra
Faces280 triangles
Edges84
Vertices14
Measures (edge length 1)
Circumradius2/2 ≈ 0.70711
Inradius14/14 ≈ 0.26726
Hypervolume2/630 ≈ 0.0022448
Diexal angleacos(–5/7) ≈ 135.58470°
Height14/7 ≈ 0.53452
Central density1
Euler characteristic2
Related polytopes
DualHepteract
ConjugateHecatonicosoctaexon
Properties
ConvexYes
OrientableYes
NatureTame

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&#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[edit | edit source]

  • Klitzing, Richard. "Zee".