# 5-polytope

 5-simplex (hexateron) 5-orthoplex, 211 (Pentacross) 5-cube (Penteract) Expanded 5-simplex Rectified 5-orthoplex 5-demicube. 121 (Demipenteract)

In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two polychoron facets. In the context of uniform 5-polytopes, a proposed name for a 5-polytope is polyteron.[1]

## 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 polychoron. 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.

## 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

## Euler characteristic

The Euler characteristic for 5-polytopes that are topological 4-spheres (including all convex 5-polytopes) is two. χ=V-E+F-C+H=2.

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 χ Symmetry (order)
5-simplex {3,3,3,3} 6 15 20 15 6 2 A5, (120)
Expanded 5-simplex t0,4{3,3,3,3} 30 120 210 180 162 2 2×A5, (240)
5-demicube {3,32,1}
h{4,3,3,3}

16 80 160 120 26 2 D5, (1920)
½BC5
5-cube {4,3,3,3} 32 80 80 40 10 2 BC5, (3820)
5-orthoplex {3,3,3,4}
{3,3,31,1}

10 40 80 80 32 2 BC5, (3840)
2×D5
Rectified 5-orthoplex t1{3,3,3,4}
t1{3,3,31,1}

40 240 400 240 42 2 BC5, (3840)
2×D5

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.

## Classification

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

• A 5-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 5-polytope is contained in the 5-polytope or its interior; otherwise, it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.
Main article: Uniform 5-polytope
• A regular 5-polytope has all identical regular polychoron facets. All regular polytera are convex.
• A prismatic 5-polytope is constructed by a Cartesian product of two lower-dimensional polytopes. A prismatic 5-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of a square and a cube), but is considered separately because it has symmetries other than those inherited from its factors.
• A 4-space tessellation is the division of four-dimensional Euclidean space into a regular grid of polychoral facets. Strictly speaking, tessellations are not polytera as they do not bound a "5D" volume, but we include them here for the sake of completeness because they are similar in many ways to polytera. A uniform 4-space tessellation is one whose vertices are related by a space group and whose facets are uniform polychora.

## Pyramids

Pyramidal 5-polytopes, or 5-pyramids, can be generated by a polychoron 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.

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
• H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Richard Klitzing, 5D, uniform polytopes (polytera)