# 7-cube

(Redirected from Hepteract)
7-cube
Rank7
TypeRegular
Notation
Bowers style acronymHept
Coxeter diagramx4o3o3o3o3o3o ()
Schläfli symbol{4,3,3,3,3,3}
Tapertopic notation1111111
Toratopic notationIIIIIII
Bracket notation[IIIIIII]
Elements
Exa14 6-cubes
Peta84 5-cubes
Tera280 tesseracts
Cells560 cubes
Faces672 squares
Edges448
Vertices128
Vertex figure6-simplex, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7}}{2}}\approx 1.32288}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Diexal angle90°
Height1
Central density1
Number of external pieces14
Level of complexity1
Related polytopes
ArmyHept
RegimentHept
Dual7-orthoplex
ConjugateNone
Abstract & topological properties
Flag count645120
Euler characteristic2
OrientableYes
Properties
SymmetryB7, order 645120
Flag orbits1
ConvexYes
Net count33064966[1]
NatureTame

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.[2] The next largest simplex that can be inscribed in a hypercube is the dodecadakon.[3]

## Vertex coordinates

The vertices of a 7-cube of edge length 1, centered at the origin, are given by:

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

## 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)

## References

1. "A091159". The On-line Encyclopedia of Integer Sequences. Retrieved 2022-12-07.
2. Adams, Joshua; Zvengrowski, Peter; Laird, Philip (2003). "Vertex Embeddings of Regular Polytopes". Expositiones Mathematicae.
3. Sloane, N. J. A. (ed.). "Sequence A019442". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.