7-orthoplex

7-orthoplex
(OFF file)
Rank	7
Type	Regular
Notation
Bowers style acronym	Zee
Coxeter diagram	o4o3o3o3o3o3x ()
Schläfli symbol	{3,3,3,3,3,4}
Bracket notation	<IIIIIII>
Elements
Exa	128 heptapeta
Peta	448 hexatera
Tera	672 pentachora
Cells	560 tetrahedra
Faces	280 triangles
Edges	84
Vertices	14
Vertex figure	Hexacontatetrapeton, edge length 1
Petrie polygons	23040 $\left\{{\dfrac {14}{1,3,5,7}}\right\}$
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {\sqrt {14}}{14}}\approx 0.26726$
Hypervolume	${\frac {\sqrt {2}}{630}}\approx 0.0022448$
Diexal angle	$\arccos \left(-{\frac {5}{7}}\right)\approx 135.58470^{\circ }$
Height	${\frac {\sqrt {14}}{7}}\approx 0.53452$
Central density	1
Number of external pieces	128
Level of complexity	1
Related polytopes
Army	Zee
Regiment	Zee
Dual	Hepteract
Conjugate	None
Abstract & topological properties
Flag count	645120
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B7, order 645120
Flag orbits	1
Convex	Yes
Net count	33064966
Nature	Tame

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:

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

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)

External links

• Klitzing, Richard. "zee".
• Wikipedia contributors. "7-orthoplex".