Graphs of three regular and three uniform 5-polytopes.
5-simplex t0.svg

5-simplex (hexateron)
5-cube t4.svg

5-orthoplex, 211
(Pentacross)
5-cube t0.svg

5-cube
(Penteract)
5-simplex t04 A4.svg

Expanded 5-simplex
5-cube t3.svg

Rectified 5-orthoplex
5-demicube t0 D5.svg

5-demicube. 121
(Demipenteract)

In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

Definition

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.

Characteristics

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]

Classification

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

Main article: Uniform 5-polytope

Main article: List_of_regular_polytopes § Convex_4

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:

Name Schläfli
symbol
(s)
Coxeter
diagram
(s)
Vertices Edges Faces Cells 4-faces Symmetry (order)
5-simplex {3,3,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 6 15 20 15 6 A5, (120)
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 80 40 10 BC5, (3820)
5-orthoplex {3,3,3,4}
{3,3,31,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
10 40 80 80 32 BC5, (3840)
2×D5

Uniform 5-polytopes

Main article: Uniform 5-polytope

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

Name Schläfli
symbol
(s)
Coxeter
diagram
(s)
Vertices Edges Faces Cells 4-faces Symmetry (order)
Expanded 5-simplex t0,4{3,3,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 30 120 210 180 162 2×A5, (240)
5-demicube {3,32,1}
h{4,3,3,3}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 80 160 120 26 D5, (1920)
½BC5
Rectified 5-orthoplex t1{3,3,3,4}
t1{3,3,31,1}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
40 240 400 240 42 BC5, (3840)
2×D5

The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png. The 5-demicube honeycomb, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube.

Pyramids

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

References

  1. ^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
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