Hepteract

Hepteract
(OFF file)
Rank	7
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Hept
Coxeter diagram	x4o3o3o3o3o3o ()
Schläfli symbol	{4,3,3,3,3,3}
Tapertopic notation	1111111
Toratopic notation	IIIIIII
Bracket notation	[IIIIIII]
Elements
Exa	14 hexeracts
Peta	84 penteracts
Tera	280 tesseracts
Cells	560 cubes
Faces	672 squares
Edges	448
Vertices	128
Vertex figure	Heptapeton, edge length √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt7}{2} ≈ 1.32288$
Inradius	$\frac12 = 0.5$
Hypervolume	1
Diexal angle	90°
Height	1
Central density	1
Number of pieces	14
Level of complexity	1
Related polytopes
Army	Hept
Regiment	Hept
Dual	Hecatonicosoctaexon
Conjugate	None
Abstract properties
Net count	33064966
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	B7, order 645120
Convex	Yes
Nature	Tame

The hepteract, or hept, also called the 7-cube, or tetradecaexon, is one of the 3 regular polyexa. It has 14 hexeracts as facets, joining 7 to a vertex. It is the 7-dimensional hypercube.

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

A regular octaexon of edge length 2 can be inscribed in the unit hepteract. The next largest simplex that can be inscribed in a hypercube is the dodecadakon.

Vertex coordinates

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

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

Representations

A hepteract has the following Coxeter diagrams:

• x4o3o3o3o3o3o (full symmetry)
• x x4o3o3o3o3o (BC6×A1 symmetry, hexxeractic prism)
• x4o x4o3o3o3o (BC5×BC2 symmetry, square-penteractic duoprism)
• x4o3o x4o3o3o (BC4×BC3 symmetry, cubic-tesseractic duoprism)
• xx4oo3oo3oo3oo3oo&#x (BC6 axial)
• oqoooooo3ooqooooo3oooqoooo3ooooqooo3oooooqoo3ooooooqo&#xt (A6 axial, vertex-first)

External links

• Klitzing, Richard. "hept".

• Wikipedia Contributors. "7-cube".