Hexeract

Hexeract
(OFF file)
Rank	6
Type	Regular
Space	Spherical
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 penteracts
Tera	60 tesseracts
Cells	160 cubes
Faces	240 squares
Edges	192
Vertices	64
Vertex figure	Hexateron, edge length √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{2} \approx 1.22475$
Edge radius	$\frac{\sqrt5}{2} \approx 1.11803$
Face radius	1
Cell radius	$\frac{\sqrt3}{2} \approx 0.86603$
Teron radius	$\frac{\sqrt2}{2} \approx 0.70711$
Inradius	$\frac12 = 0.5$
Hypervolume	1
Dipetal angle	90°
Height	1
Central density	1
Number of pieces	12
Level of complexity	1
Related polytopes
Army	Ax
Regiment	Ax
Dual	Hexacontatetrapeton
Conjugate	None
Abstract properties
Flag count	46080
Net count	502110
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B6, order 46080
Convex	Yes
Nature	Tame

The hexeract, or ax, also called the 6-cube or dodecapeton, is one of the 3 regular polypeta. It has 12 penteracts as facets, joining 6 to a vertex. It is the 6-dimensional hypercube.

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

It can be alternated into a demihexeract, which is uniform.

Vertex coordinates

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

• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12\right).$

Representations

A hexeract has the following Coxeter diagrams:

• x4o3o3o3o3o (full symmetry)
• x4o3o x4o3o () (B₃×B₃ symmetry, cubic duoprism)
• x4o x4o3o3o () (B₄×B₂ symmetry, square-tesseractic duoprism)
• x x4o3o3o3o () (B₅×B₁ symmetry, penteractic prism)
• x4o x4o x4o () (B₂×B₂×B₂ symmetry, square trioprism)
• x x4o x4o3o () (B₃×B₃×A₁ symmetry, square-cubic duoprismatic prism)
• x x x4o3o3o () (B₄×A₁×A₁ symmetry, tesseract prismatic prism)
• x x x4o x4o () (B₂×B₂×A₁×A₁ symmetry, square duoprismatic prismatic prism)
• x x x x4o3o () (B₃×A₁×A₁×A₁ symmetry, cubic prism prism prism)
• x x x x x4o () (five dimensions different)
• x x x x x x () (all six dimensions different)
• xx4oo3oo3oo3oo&#x (B₅ axial, penteract 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".