6-cube

6-cube
(OFF file)
Rank	6
Type	Regular
Notation
Bowers style acronym	Ax
Coxeter diagram	x4o3o3o3o3o ()
Schläfli symbol	{4,3,3,3,3}
Tapertopic notation	111111
Toratopic notation	IIIIII
Bracket notation	[IIIIII]
Elements
Peta	12 5-cubes
Tera	60 tesseracts
Cells	160 cubes
Faces	240 squares
Edges	192
Vertices	64
Vertex figure	5-simplex, edge length √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {6}}{2}}\approx 1.22475$
Edge radius	${\frac {\sqrt {5}}{2}}\approx 1.11803$
Face radius	1
Cell radius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Teron radius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {1}{2}}=0.5$
Hypervolume	1
Dipetal angle	90°
Height	1
Central density	1
Number of external pieces	12
Level of complexity	1
Related polytopes
Army	Ax
Regiment	Ax
Dual	6-orthoplex
κ ?	Alternative 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-cube (OBSA: ax), also called the hexeract or dodecapeton, is a regular 6-polytope. It has twelve 5-cubes as facets, joining 6 to a vertex. It is the 6-dimensional hypercube.

It is also a 5-cubic prism, a cubic duoprism, square trioprism, dodecahedron-small stellated dodecahedral gyropeton, and 12-3-5 gyropeton.

It can be alternated into a 6-demicube, which is uniform.

Vertex coordinates

The vertices of a hexeract of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$.

Representations

A 6-cube has the following Coxeter diagrams:

• x4o3o3o3o3o () (full symmetry)
• x4o3o2x4o3o () (B₃×B₃ symmetry, cubic duoprism)
• x4o2x4o3o3o () (B₄×B₂ symmetry, square-tesseractic duoprism)
• x2x4o3o3o3o () (B₅×B₁ symmetry, 5-cube prism)
• x4o2x4o2x4o () (B₂×B₂×B₂ symmetry, square trioprism)
• x2x4o2x4o3o () (B₃×B₃×A₁ symmetry, square-cubic duoprismatic prism)
• x2x2x4o3o3o () (B₄×K₂ symmetry, tesseract prismatic prism)
• x2x2x4o2x4o () (B₂×B₂×K₂ symmetry, square duoprismatic prismatic prism)
• x2x2x2x4o3o () (B₃×K₃ symmetry, cubic prism prism prism)
• x2x2x2x2x4o () (B₂×K₄ symmetry, five dimensions different)
• x2x2x2x2x2x () (K₆ symmetry, all six dimensions different)
• xx4oo3oo3oo3oo&#x (B₅ axial, 5-cube prism)
• qo3oo3oq *b3oo3oo3oo&#zx (D₄ symmetry, hull of two demihexeracts)
• oqooooo3ooqoooo3oooqooo3ooooqoo3oooooqo&#xt (A₅ axial, vertex-first)

Gallery

• 
  A hexeract projected into a rhombic triacontahedron envelope, this projection is related to icosahedral quasicrystals.

External links

• Bowers, Jonathan. "Category 1: Primary Polypeta" (#2).

• Klitzing, Richard. "ax".
• Wikipedia contributors. "6-cube".
• Hi.gher.Space Wiki Contributors. "Geopeton".