# 5-cube

5-cube
Rank5
TypeRegular
Notation
Bowers style acronymPent
Coxeter diagramx4o3o3o3o ()
Schläfli symbol{4,3,3,3}
Tapertopic notation11111
Toratopic notationIIIII
Bracket notation[IIIII]
Elements
Tera10 tesseracts
Cells40 cubes
Faces80 squares
Edges80
Vertices32
Vertex figurePentachoron, edge length 2
Petrie polygons16 5D decagons
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5}}{2}}\approx 1.11803}$
Face radius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Cell radius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70111}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Diteral angle90°
Height1
Central density1
Number of external pieces10
Level of complexity1
Related polytopes
ArmyPent
RegimentPent
Dual5-orthoplex
ConjugateNone
Abstract & topological properties
Flag count3840
Euler characteristic2
OrientableYes
SkeletonQ 5
Properties
SymmetryB5, order 3840
Flag orbits1
ConvexYes
Net count9694[1]
NatureTame

The 5-cube, also called the decateron, penteract, or pent, is a regular 5-polytope. 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, square-cube duoprism, and even a cubic prismatic prism along with many other symmetries.

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

It can be alternated into a 5-demicube, which is uniform, but not regular.

## Vertex coordinates

The vertices of a 5-cube of edge length 1, centered at the origin, are given by:

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

## Representations

A 5-cube has the following Coxeter diagrams:

• x4o3o3o3o () (full symmetry)
• x x4o3o3o () (B4×A1 symmetry, tesseractic prism)
• x4o x4o3o () (B3×B2 symmetry, square-cubic duoprism)
• x x x4o3o () (B3×K2 symmetry, cubic prismatic prism)
• x x4o x4o () (B2×B2×A1 symmetry, square duoprismatic prism)
• x x x x4o () (B2×K3 symmetry, square prismatic prismatic prism)
• x x x x x () (K5 symmetry, all five dimensions separate)
• xx4oo3oo3oo&#x (B4 axial, tesseract atop tesseract)
• xx xx4oo3oo&#x (B3×A1 axial, cubic prism bases)
• xx4oo xx4oo&#x (B3×B2 axial, square duoprismatic bases)
• xx xx xx4oo&#x (B2×K2 axial, square prismatic prism bases)
• xx xx xx xx&#x (K4 symmetry, bases have four separate dimensions)
• oqo xxx4ooo3ooo&#xt (B3×A1 axial, cell-first)
• xxxx4oooo oqoo3ooqo&#xt (B2×A1 axial, face-first)
• xxxxx oqooo3ooqoo3oooqo&#xt (A3×A1 axial, edge-first)
• oqoooo3ooqooo3oooqoo3ooooqo&#xt (A4 axial, vertex-first)
• qo3oo3oq *b3oo3oo&#zx (D5 symmetry)