# 7-orthoplex

7-orthoplex
Rank7
TypeRegular
Notation
Bowers style acronymZee
Coxeter diagramo4o3o3o3o3o3x ()
Schläfli symbol{3,3,3,3,3,4}
Bracket notation<IIIIIII>
Elements
Exa128 6-simplices
Peta448 5-simplices
Tera672 pentachora
Cells560 tetrahedra
Faces280 triangles
Edges84
Vertices14
Vertex figure6-orthoplex, edge length 1
Petrie polygons23040 ${\displaystyle \left\{{\dfrac {14}{1,3,5,7}}\right\}}$
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {\sqrt {14}}{14}}\approx 0.26726}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{630}}\approx 0.0022448}$
Diexal angle${\displaystyle \arccos \left(-{\frac {5}{7}}\right)\approx 135.58470^{\circ }}$
Height${\displaystyle {\frac {\sqrt {14}}{7}}\approx 0.53452}$
Central density1
Number of external pieces128
Level of complexity1
Related polytopes
ArmyZee
RegimentZee
Dual7-cube
ConjugateNone
Abstract & topological properties
Flag count645120
Euler characteristic2
OrientableYes
Properties
SymmetryB7, order 645120
Flag orbits1
ConvexYes
Net count33064966
NatureTame

The 7-orthoplex (also called the 7-cross-polytope, heptacross, hecatonicosoctaexon, or zee) is a regular 7-polytope. It has 128 regular 6-simplices 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 7-segmentotope, as a 6-simplicial antiprism.

## Vertex coordinates

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

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

## Representations

A hecatonicosoctaexon has the following Coxeter diagrams:

• o4o3o3o3o3o3x () (full symmetry)
• o3o3o3o3o3x *b3o () (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)