5-polytope

Graphs of three regular and one uniform polytopes.

5-simplex (hexateron)	5-orthoplex, 211 (Pentacross)
5-cube (Penteract)	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. A proposed name for 5-polytopes is polyteron.

Definition

A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and hypercells. 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 hypercell is a polychoron. Furthermore, the following requirements must be met:

1. Each cell must join exactly two hypercells.
2. Adjacent hypercells 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} - Hexateron (5-simplex)
2. {4,3,3,3} - Penteract (5-hypercube)
3. {3,3,3,4} - Pentacross (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 two semiregular 5-polytope, their elements are:

Name	Schläfli symbol	Vertices	Edges	Faces	Cells	4-faces	χ
5-simplex	{3,3,3,3}	6	15	20	15	6	2
5-orthoplex	{3,3,3,4}	10	40	80	80	32	2
5-demicube	{31,2,1}	16	80	160	120	26	2
5-cube	{4,3,3,3}	32	80	80	40	10	2
Rectified pentacross	t1{3,3,3,4}	40	240	400	240	42	2

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.

• A uniform 5-polytope has a symmetry group under which all vertices are equivalent, and its facets are uniform polychora. The edges of a uniform 5-polytope must be equal in length.

Main article: Uniform 5-polytope

• A semi-regular 5-polytope contains two or more types of regular polychoral facets. There is only one such figure, called a demipenteract.

• A regular 5-polytope has all identical regular polychoron facets. All regular polytera are convex.

Main article: List_of_regular_polytopes#Convex_4

• A prismatic 5-polytope is constructed by a the 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 squares 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.

See also

• List of regular polytopes#Five-dimensional regular polytopes

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, editied 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)

External links

• Polytopes of Various Dimensions, Jonathan Bowers
• Uniform Polytera, Jonathan Bowers
• Olshevsky, George, Polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary, Garrett Jones