6-orthoplex

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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
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Cell radius
Teron radius
Inradius
Hypervolume
Dipetal angle
Height
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[edit | edit source]

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

  • .

Representations[edit | edit source]

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[edit | edit source]

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.

External links[edit | edit source]