# Hecatonicosoctaexon

(Redirected from 7-orthoplex)
Hecatonicosoctaexon
Rank7
TypeRegular
SpaceSpherical
Notation
Bowers style acronymZee
Coxeter diagramo4o3o3o3o3o3x ()
Schläfli symbol{3,3,3,3,3,4}
Bracket notation<IIIIIII>
Elements
Exa128 heptapeta
Peta448 hexatera
Tera672 pentachora
Cells560 tetrahedra
Faces280 triangles
Edges84
Vertices14
Vertex figureHexacontatetrapeton, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2} \approx 0.70711}$
Inradius${\displaystyle \frac{\sqrt{14}}{14} \approx 0.26726}$
Hypervolume${\displaystyle \frac{\sqrt2}{630} \approx 0.0022448}$
Diexal angle${\displaystyle \arccos\left(-\frac57\right) \approx 135.58470^\circ}$
Height${\displaystyle \frac{\sqrt{14}}{7} \approx 0.53452}$
Central density1
Number of external pieces128
Level of complexity1
Related polytopes
ArmyZee
RegimentZee
DualHepteract
ConjugateNone
Abstract & topological properties
Flag count645120
Euler characteristic2
OrientableYes
Properties
SymmetryB7, order 645120
ConvexYes
Net count33064966
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

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

• ${\displaystyle \left(\pm\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0\right).}$

## Representations

A hecatonicosoctaexon has the following Coxeter diagrams:

• o4o3o3o3o3o3x (full symmetry)
• o3o3o *b3o3o3o3x () (D7 symmetry)
• xo3oo3oo3oo3oo3ox&#x (A6 axial, heptapetal antiprism)
• ooo4ooo3ooo3ooo3ooo3oxo&#xt (B6 axial, hexacontatetrapetal bipyramid)
• qo oo4oo3oo3oo3oo3ox&#zx (B6×A1 symmetry)
• xox ooo4ooo3ooo3ooo3oxo&#xt (B5×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 (B4×A2 axial, triangle-first)
• xoo3oox oxo3ooo3ooo *d3ooo&#xt (D4×2 symmetry, triangle-first with half symmetry)
• oxo4oooo xoo3ooo3ooo3oox&#xt (A4×B2 axial, pentachoron-first)
• oxo oxo xoo3ooo3ooo3oox&#xt (A4×A1×A1 symmetry, pentachoron-first with half symmetry)
• xo4oo oo4oo3oo3oo3ox&#zx (B5×B2 symmetry, square-triacontaditeral duotegum)
• xoo3ooo3oox ooo4ooo3oxo&#xt (B3×A3 axial, tetrahedron-first)
• xoo3ooo3oox ooo3oxo3ooo&#xt (A3×A3 axial, tetrahedron-first with half symmetry)
• oo4oo3xo oo4oo3oo3ox&#zx (B4×B3 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×B2 symmetry)
• oxooo3oooxo ooxoo3ooooo4ooooo&#xcr (B3×A2 symmetry)
• o(xoo)o o(xoo)o o(oxo)o o(oxo)o o(oox)o o(oox)o&#xt (square triotegum bipyramid)