# Penteract

(Redirected from 5-hypercube)
Penteract
Rank5
TypeRegular
SpaceSpherical
Bowers style acronymPent
Info
Coxeter diagramx4o3o3o3o
Schläfli symbol{4,3,3,3}
Tapertopic notation11111
Toratopic notationIIIII
Bracket notation[IIIII]
SymmetryBC5, order 3840
ArmyPent
RegimentPent
Elements
Vertex figurePentachoron, edge length 2
Tera10 tesseracts
Cells40 cubes
Faces80 squares
Edges80
Vertices32
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt5}{2} ≈ 1.11803}$
Face radius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Cell radius${\displaystyle \frac{\sqrt2}{2} ≈ 0.70111}$
Inradius${\displaystyle \frac12 = 0.5}$
Hypervolume1
Diteral angle90°
Height1
Central density1
Euler characteristic2
Number of pieces10
Level of complexity1
Related polytopes
ConjugatePenteract
Properties
ConvexYes
OrientableYes
NatureTame

The penteract, or pent, also called the 5-cube or decateron, is one of the 3 regular polytera. It has 10 tesseracts as facets, joining 5 to a vertex. It is the 5-dimensional hypercube. As such, it is also a tesseractic prism and square-cube duoprism.

Like the hypercubes of every other dimension, the penteract can tile 5D Euclidean space in the penteractic pentacomb.

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

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).}$

## Representations

A penteract has the following Coxeter diagrams:

• x4o3o3o3o (full ysmmetry)
• x x4o3o3o (BC4×A1 symmetry, tesseractic prism)
• x4o x4o3o (BC3×BC2 symmetry, square-cubic duoprism)
• x x x4o3o (BC3×A1×A1 symmetry, cubic prismatic prism)
• x x4o x4o (BC2×BC2×A1 symmetry, square duoprismatic prism)
• x x x x4o (BC2×A1×A1×A1 symmetry, square prismatic prismatic prism)
• x x x x x (A1×A1×A1×A1×A1 symmetry, all five dimensions separate)
• xx4oo3oo3oo&#x (BC4 axial, tesseract atop tesseract)
• xx xx4oo3oo&#x (BC3×A1 axial, cubic prism bases)
• xx4oo xx4oo&#x (BC2×BC2 axial, square duoprismatic bases)
• xx xx xx4oo&#x (BC2×A1×A1 axial, square prismatic prism bases)
• xx xx xx xx&#x (A1×A1×A1×A1 symetry, bases have four separate dimensions)
• oqo xxx4ooo3ooo&#xt (BC3×A1 axial, cell-first)
• xxxx4oooo oqoo3ooqo&#xt (BC2×A1 axial, face-first)
• xxxxx oqooo3ooqoo3oooqo&#xt (A3×A1 axial, edge-first)
• oqoooo3ooqooo3oooqoo3ooooqo&#xt (A4 axial, vertex-first)
• qo3oo3oq *b3oo3oo&#zx (D5 symmetry)

## Related polytopes

o4o3o3o3o truncations
Name OBSA CD diagram Picture
Penteract pent
Rectified penteract rin
Truncated penteract tan
Bitruncated penteract bittin
Small rhombated penteract sirn
Small birhombated penteract sibrant
Great rhombated penteract girn
Great birhombated penteract gibrant
Small prismated penteract span
Prismatotruncated penteract pattin