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penteract (pent)
Type uniform 5-polytope
Schläfli symbol {4,3,3,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-faces 10 tesseracts
Cells 40 cubes
Faces 80 squares
Edges 80
Vertices 32
Vertex figure 5-cube verf.png
Coxeter group B5, [4,33], order 3840
Dual 5-orthoplex
Base point (1,1,1,1,1,1)
Circumradius sqrt(5)/2 = 1.118034
Properties convex, isogonal regular, Hanner polytope

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

It is represented by Schläfli symbol {4,3,3,3} or {4,33}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word pénte, for 'five' (dimensions), and the word tesseract (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets.

Related polytopes[edit]

It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes.

Applying an alternation operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.

The 5-cube can be seen as an order-3 tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb.

As a configuration[edit]

This configuration matrix represents the 5-cube. The rows and columns correspond to vertices, edges, faces, cells, and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

Cartesian coordinates[edit]

The Cartesian coordinates of the vertices of a 5-cube centered at the origin and having edge length 2 are


while this 5-cube's interior consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1 for all i.


n-cube Coxeter plane projections in the Bk Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.

Orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t0.svg 4-cube t0.svg 5-cube t0 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane Other B2 A3
Graph 5-cube column graph.svg 5-cube t0 B2.svg 5-cube t0 A3.svg
Dihedral symmetry [2] [4] [4]
More orthographic projections
2d of 5d 3.svg
Wireframe skew direction
B5 Coxeter plane
Penteract graph.svg
Vertex-edge graph.
Perspective projections
Penteract projected.png
A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D.
The Net of 5-cube.png
4D net of the 5-cube, perspective projected into 3D.


The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a pentagonal antiprism. The 80 edges project into 40 external edges and 40 internal ones. The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron. The projection vectors are u = {1, φ, 0, -1, φ}, v = {φ, 0, 1, φ, 0}, w = {0, 1, φ, 0, -1}, where φ is the golden ratio, .

rhombic icosahedron 5-cube
Perspective orthogonal
Rhombic icosahedron.png Dual dodecahedron t1 H3.png 5-cube t0.svg


The 5-cube has Coxeter group symmetry B5, abstract structure , order 3840, containing 25 hyperplanes of reflection. The Schläfli symbol for the 5-cube, {4,3,3,3}, matches the Coxeter notation symmetry [4,3,3,3].


All hypercube have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-orthotope, { }5, and upwards as orthogonal edges are constrained to be of equal length. The vertices in a prism are equal to the product of the vertices in the elements. The edges of a prism can be partitioned into the number of edges in an element times the number of vertices in all the other elements.

Description Schläfli symbol Coxeter-Dynkin diagram Vertices Edges Coxeter notation
5-cube {4,3,3,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 32 80 [4,3,3,3] 3840
tesseractic prism {4,3,3}×{ } CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png 16×2 = 32 64 + 16 = 80 [4,3,3,2] 768
cube-square duoprism {4,3}×{4} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png 8×4 = 32 48 + 32 = 80 [4,3,2,4] 384
cube-rectangle duoprism {4,3}×{ }2 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png 8×22 = 32 48 + 2×16 = 80 [4,3,2,2] 192
square-square duoprism prism {4}2×{ } CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png 42×2 = 32 2×32 + 16 = 80 [4,2,4,2] 128
square-rectangular parallelepiped duoprism {4}×{ }3 CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png 4×23 = 32 32 + 3×16 = 80 [4,2,2,2] 64
5-orthotope { }5 CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png 25 = 32 5×16 = 80 [2,2,2,2] 32

Related polytopes[edit]

The 5-cube is 5th in a series of hypercube:

Petrie polygon orthographic projections
1-simplex t0.svg 2-cube.svg 3-cube graph.svg 4-cube graph.svg 5-cube graph.svg 6-cube graph.svg 7-cube graph.svg 8-cube.svg
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube

The regular skew polyhedron {4,5| 4} can be realized within the 5-cube, with its 32 vertices, 80 edges, and 40 square faces, and the other 40 square faces of the 5-cube become square holes.

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

B5 polytopes
5-cube t4.svg
5-cube t3.svg
5-cube t2.svg
5-cube t1.svg
5-cube t0.svg
5-cube t34.svg
5-cube t24.svg
5-cube t23.svg
5-cube t14.svg
5-cube t13.svg
5-cube t12.svg
5-cube t04.svg
5-cube t03.svg
5-cube t02.svg
5-cube t01.svg
5-cube t234.svg
5-cube t134.svg
5-cube t124.svg
5-cube t123.svg
5-cube t034.svg
5-cube t024.svg
5-cube t023.svg
5-cube t014.svg
5-cube t013.svg
5-cube t012.svg
5-cube t1234.svg
5-cube t0234.svg
5-cube t0134.svg
5-cube t0124.svg
5-cube t0123.svg
5-cube t01234.svg


  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Klitzing, Richard. "5D uniform polytopes (polytera) o3o3o3o4x - pent".

External links[edit]

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds