# Hepteract

(Redirected from 7-hypercube)
Hepteract
Rank7
TypeRegular
SpaceSpherical
Bowers style acronymHept
Info
Coxeter diagramx4o3o3o3o3o3o
Schläfli symbol{4,3,3,3,3,3}
SymmetryBC7, order 645120
ArmyHept
RegimentHept
Elements
Vertex figureHeptapeton, edge length 2
Exa14 hexeracts
Peta84 penteracts
Tera280 tesseracts
Cells560 cubes
Faces672 squares
Edges448
Vertices128
Measures (edge length 1)
Hypervolume1
Diexal angle90°
Central density1
Euler characteristic2
Related polytopes
DualHecatonicosoctaexon
ConjugateHepteract
Properties
ConvexYes
OrientableYes
NatureTame

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.[1] The next largest simplex that can be inscribed in a hypercube is the dodecadakon.[2]

## Vertex coordinates

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

• (±1/2, ±1/2, ±1/2, ±1/2, ±1/2, ±1/2, ±1/2).

## 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)

## References

1. Adams, Joshua; Zvengrowski, Peter; Laird, Philip (2003). "Vertex Embeddings of Regular Polytopes". Expositiones Mathematicae.
2. Sloane, N. J. A. (ed.). "Sequence A019442". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.