6-orthoplex

6-orthoplex
(OFF file)
Rank	6
Type	Regular
Notation
Bowers style acronym	Gee
Coxeter diagram	x3o3o3o3o4o ()
Schläfli symbol	{3,3,3,3,4}
Bracket notation	<IIIIII>
Elements
Peta	64 5-simplices
Tera	192 pentachora
Cells	240 tetrahedra
Faces	160 triangles
Edges	60
Vertices	12
Vertex figure	5-orthoplex, edge length 1
Petrie polygons	1920 $\left\{{\frac {12}{1,3,5}}\right\}$
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Edge radius	${\frac {1}{2}}=0.5$
Face radius	${\frac {\sqrt {6}}{6}}\approx 0.40825$
Cell radius	${\frac {\sqrt {2}}{4}}\approx 0.35355$
Teron radius	${\frac {\sqrt {10}}{10}}\approx 0.31623$
Inradius	${\frac {\sqrt {3}}{6}}\approx 0.28868$
Hypervolume	${\frac {1}{90}}\approx 0.011111$
Dipetal angle	$\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
Height	${\frac {\sqrt {3}}{3}}\approx 0.57735$
Central density	1
Number of external pieces	64
Level of complexity	1
Related polytopes
Army	Gee
Regiment	Gee
Dual	6-cube
Conjugate	None
Abstract & topological properties
Flag count	46080
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B6, order 46080
Flag orbits	1
Convex	Yes
Net count	502110
Nature	Tame

The 6-orthoplex (OBSA: 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:

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

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.

External links

• Bowers, Jonathan. "Category 1: Primary Polypeta" (#3).

• Klitzing, Richard. "gee".
• Wikipedia contributors. "6-orthoplex".
• Hi.gher.Space Wiki Contributors. "Aeropeton".