Hepteract

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Hepteract
7-cube graph.svg
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)
Circumradius7/2 ≈ 1.32288
Inradius1/2 = 0.5
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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

  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.

External links[edit | edit source]