# 6-orthoplex

6-orthoplex
Rank6
TypeRegular
Notation
Bowers style acronymGee
Coxeter diagramx3o3o3o3o4o ()
Schläfli symbol{3,3,3,3,4}
Bracket notation<IIIIII>
Elements
Peta64 5-simplices
Tera192 pentachora
Cells240 tetrahedra
Faces160 triangles
Edges60
Vertices12
Vertex figure5-orthoplex, edge length 1
Petrie polygons1920 ${\displaystyle \left\{{\frac {12}{1,3,5}}\right\}}$
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Edge radius${\displaystyle {\frac {1}{2}}=0.5}$
Face radius${\displaystyle {\frac {\sqrt {6}}{6}}\approx 0.40825}$
Cell radius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35355}$
Teron radius${\displaystyle {\frac {\sqrt {10}}{10}}\approx 0.31623}$
Inradius${\displaystyle {\frac {\sqrt {3}}{6}}\approx 0.28868}$
Hypervolume${\displaystyle {\frac {1}{90}}\approx 0.011111}$
Dipetal angle${\displaystyle \arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }}$
Height${\displaystyle {\frac {\sqrt {3}}{3}}\approx 0.57735}$
Central density1
Number of external pieces64
Level of complexity1
Related polytopes
ArmyGee
RegimentGee
Dual6-cube
ConjugateNone
Abstract & topological properties
Flag count46080
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
ConvexYes
Net count502110
NatureTame

The 6-orthoplex, or gee, also called the hexacross or hexacontatetrapeton, is a regular 6-polytope. It has 64 regular 5-simplices as facets, joining 3 to a tetrahedron peak and 32 to a vertex in a triacontaditeral arrangement. It is the 6-dimensional orthoplex. It is also an octahedral duotegum and square triotegum, triacontaditeric tegum, icosahedron-great icosahedron step prism, and 12-3-5 step prism.

It can also be seen as a segmentopeton as a 5-simplex antiprism.

## Vertex coordinates

The vertices of a regular 6-orthoplex of edge length 1, centered at the origin, are given by all permutations of:

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

## Representations

A 6-orthoplex has the following Coxeter diagrams:

• x3o3o3o3o4o () (full symmetry)
• x3o3o3o3o *d3o () (D6 symmetry)
• xo3oo3oo3oo3ox&#x (A5 axial, hexateric antiprism)
• ooo4ooo3ooo3ooo3oxo&#xt (B5 axial, triacontaditeric bipyramid)
• qo oo4oo3oo3oo3ox&#zx (B5×A1 symmetry)
• oo3ooo3ooo *b3ooo3oxo&#xt (D5 axial, still triacontaditeric bipyramid)
• qo oo3oo3oo *c3oo3ox&#zx (D5×A1 symmety)
• oxoo3oooo3oooo3ooox&#x (A4 axial)
• oqo xoo3ooo3ooo3oox&#xt (A4×A1 axial, pentachoron-first)
• xox ooo4ooo3ooo3oxo&#xt (B4×A1 symmetry, edge-first)
• xox oxo3ooo3ooo *c3ooo&#xt (D4×A1 axial, still edge-first)
• xo4oo oo4oo3oo3ox&#zx (B4×B2 symmetry, square-hexadecachoron duotegum)
• xo xo ox3oo3oo *d3oo&#zx (D4×A1×A1 symmetry, rectangle-demitesseract duotegum)
• oxo4ooo xoo3ooo3oox&#xt (A3×B2 symmetry, tetrahedron-first)
• oxo oxo xoo3ooo3oox&#xt (A3×A1×A1 symmetry, still tetrahedron-first)
• xoxo oxoo3oooo3ooox&#xr (A3×A1 axial)
• xoo3oox ooo4ooo3oox&#xt (B3×A2 axial, triangle-first)
• xoo3oox ooo3oxo3ooo&#xt (A3×A2 axial, triangle-first)
• oo4oo3xo oo4oo3ox&#zx (B3×B3 symmetry, octahedral duotegum)
• oo3xo3oo oo3ox3oo&#zx (A3×A3 symmetry, tetratetrahedral duotegum)
• xooo3ooxo oxoo3ooox&#xr (A2×A2 symmetry)
• xoo4ooo oxo4ooo oox4ooo&#zx (B2×B2×B2 symetry, square triotegum)
• xoo xoo oxo oxo oox oox&#zx (rectangular triotegum)

## Related polytopes

The regiment of the 6-orthoplex includes a total of 13 known uniform members, including itself, 1 with D6 symmetry (the triacontadihemihexeract), 4 with hexateric antiprism symmetry, 2 with doubled icosahedral step prism symmetry, 1 with icosahedral step prism symmetry, and 4 with triangular disphenoidal antiprismatic symmetry. The regiment also includes a number of scaliforms.