Graphs of three regular and three uniform polytopes.

5-simplex (hexateron)

5-orthoplex, 211


Expanded 5-simplex

Rectified 5-orthoplex

5-demicube. 121

In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.


A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met:

  1. Each cell must join exactly two 4-faces.
  2. Adjacent 4-faces are not in the same four-dimensional hyperplane.
  3. The figure is not a compound of other figures which meet the requirements.


The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.[1]

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]


5-polytopes may be classified based on properties like "convexity" and "symmetry".

Main article: Uniform 5-polytope

Regular 5-polytopes

Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.

There are exactly three such convex regular 5-polytopes:

  1. {3,3,3,3} - 5-simplex
  2. {4,3,3,3} - 5-cube
  3. {3,3,3,4} - 5-orthoplex

For the 3 convex regular 5-polytopes and three semiregular 5-polytope, their elements are:

VerticesEdgesFacesCells4-facesSymmetry (order)
5-simplex{3,3,3,3}61520156A5, (120)
5-cube{4,3,3,3}3280804010BC5, (3820)

1040808032BC5, (3840)

Uniform 5-polytopes

Main article: Uniform 5-polytope

For three of the semiregular 5-polytope, their elements are:

VerticesEdgesFacesCells4-facesSymmetry (order)
Expanded 5-simplext0,4{3,3,3,3}301202101801622×A5, (240)

168016012026D5, (1920)
Rectified 5-orthoplext1{3,3,3,4}

4024040024042BC5, (3840)

The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube.


Pyramidal 5-polytopes, or 5-pyramids, can be generated by a 4-polytope base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.

See also


  1. 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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
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