7-cube

7-cube
(OFF file)
Rank	7
Type	Regular
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 6-cubes
Peta	84 5-cubes
Tera	280 tesseracts
Cells	560 cubes
Faces	672 squares
Edges	448
Vertices	128
Vertex figure	6-simplex, edge length √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {7}}{2}}\approx 1.32288$
Inradius	${\frac {1}{2}}=0.5$
Hypervolume	1
Diexal angle	90°
Height	1
Central density	1
Number of external pieces	14
Level of complexity	1
Related polytopes
Army	Hept
Regiment	Hept
Dual	7-orthoplex
Conjugate	None
Abstract & topological properties
Flag count	645120
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B7, order 645120
Flag orbits	1
Convex	Yes
Net count	33064966
Nature	Tame

The hepteract (OBSA: hept) also called the 7-cube, or tetradecaexon, is one of the 3 convex regular 7-polytopes. It has 14 6-cubes 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 7-simplex 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 7-cube of edge length 1, centered at the origin, are given by:

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

Representations

A 7-cube has the following Coxeter diagrams:

• x4o3o3o3o3o3o () (full symmetry)
• x x4o3o3o3o3o () (B₆×A₁ symmetry, hexeractic prism)
• x4o x4o3o3o3o () (B₅×B₂ symmetry, square-penteractic duoprism)
• x4o3o x4o3o3o () (B₄×B₃ symmetry, cubic-tesseractic duoprism)
• xx4oo3oo3oo3oo3oo&#x (B₆ axial)
• oqoooooo3ooqooooo3oooqoooo3ooooqooo3oooooqoo3ooooooqo&#xt (A₆ axial, vertex-first)

External links

• Klitzing, Richard. "hept".
• Wikipedia contributors. "7-cube".