Penteract

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Penteract
5-cube t0.svg
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
Edge radius1
Face radius
Cell radius
Inradius
Hypervolume1
Diteral angle90°
Height1
Central density1
Euler characteristic2
Number of pieces10
Level of complexity1
Related polytopes
DualTriacontaditeron
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[edit | edit source]

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

Representations[edit | edit source]

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[edit | edit source]

o4o3o3o3o truncations
Name OBSA CD diagram Picture
Penteract pent CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t0.svg
Rectified penteract rin CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t1.svg
Penteractitriacontaditeron nit CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t2.svg
Rectified triacontaditeron rat CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t3.svg
Triacontaditeron tac CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t4.svg
Truncated penteract tan CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t01.svg
Bitruncated penteract bittin CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t12.svg
Bitruncated triacontaditeron bittit CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t23.svg
Truncated triacontaditeron tot CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t34.svg
Small rhombated penteract sirn CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t02.svg
Small birhombated penteract sibrant CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t13.svg
Small rhombated triacontaditeron sart CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t24.svg
Great rhombated penteract girn CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t012.svg
Great birhombated penteract gibrant CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t123.svg
Great rhombated triacontaditeron gart CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t234.svg
Small prismated penteract span CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t03.svg
Small prismated triacontaditeron spat CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t14.svg
Prismatotruncated penteract pattin CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t013.svg
Prismatorhombated triacontaditeron pirt CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t124.svg
Prismatorhombated penteract prin CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t023.svg
Prismatotruncated triacontaditeron pattit CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t134.svg
Great prismated penteract gippin CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t0123.svg
Great prismated triacontaditeron gippit CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t1234.svg
Small cellated penteract scant CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t04.svg
Celliprismated triacontaditeron capt CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t014.svg
Cellirhombated penteract carnit CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t024.svg
Celligreatorhombated penteract cogrin CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t0124.svg
Celliprismated penteract cappin CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t034.svg
Celliprismatotruncated penteract captint CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t0134.svg
Celligreatorhombated triacontaditeron cogart CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t0234.svg
Great cellated penteract gacnet CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t01234.svg

External links[edit | edit source]