# 5-cube

(Redirected from Penteract)
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 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 5-demicube, which is uniform.

## 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)