5-cube

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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
Edge radius1
Face radius
Cell radius
Inradius
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[edit | edit source]

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

  • .

Representations[edit | edit source]

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)

Gallery[edit | edit source]

External links[edit | edit source]


References[edit | edit source]

  1. "A091159". The On-line Encyclopedia of Integer Sequences. Retrieved 2022-12-07.