6-polytope

Graphs of three regular and three uniform polytopes

6-simplex (Heptapeton)	6-orthoplex, 311
6-cube (Hexeract)	6-demicube 131 (Demihexeract)
221	122

In six-dimensional geometry, a 6-polytope is a polytope, bounded by 5-polytope facets.

In the context of uniform polytopes, a proposed name polypeton (plural: polypeta) has been advocated, from the Greek root poly- meaning "many", a shortened penta- meaning "five", and suffix -on. "Five" refers to the dimension of the 5-polytope facets.

Definition

A 6-polytope, or polypeton, is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six 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. A 4-face is a polychoron, and a 5-face is a polyteron. Furthermore, the following requirements must be met:

• Each 4-face must join exactly two 5-faces (facets).
• Adjacent facets are not in the same five-dimensional hyperplane.
• The figure is not a compound of other figures which meet the requirements.

Regular 6-polytopes

Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol {p,q,r,s,t} with t {p,q,r,s} polyteron facets around each cell.

There are only three such convex regular 6-polytopes:

• {3,3,3,3,3} - 6-simplex
• {4,3,3,3,3} - 6-cube
• {3,3,3,3,4} - 6-orthoplex

There are no nonconvex regular polytopes of 5 or more dimensions.

Euler characteristic

The Euler characteristic for 6-polytopes that are topological 5-spheres (including all convex 6-polytopes) is zero: χ=V-E+F-C+f₄-f₅=0.

For the 3 convex regular 6-polytopes, their elements are:

Name	Schläfli symbol	Coxeter-Dynkin	Vertices	Edges	Faces	Cells	4-faces	5-faces	χ
6-simplex	{3,3,3,3,3}		7	21	35	35	21	7	0
6-orthoplex	{3,3,3,3,4}		12	60	160	240	192	64	0
6-cube	{4,3,3,3,3}		64	192	240	160	60	12	0

There are three more simple convex 6-polytopes, uniform polytopes with Coxeter-Dynkin diagrams with one end-node ringed.

Name	Schläfli symbol	Coxeter-Dynkin	Vertices	Edges	Faces	Cells	4-faces	5-faces	χ
6-orthoplex (alternate construction)	{33,1,1}		12	60	160	240	192	64	0
6-demicube	{31,3,1}		32	240	640	640	252	44	0
221 polytope	{32,2,1}		27	216	720	1080	648	99	0
122 polytope	{31,2,2}		72	720	2160	2160	702	54	0

Classification

6-polytopes may be classified by properties like "convexity" and "symmetry".

• A 6-polytope is convex if its boundary (including its 5-faces, 4-faces, cells, faces and edges) does not intersect itself and the line segment joining any two points of the 6-polytope is contained in the polypeton or its interior; otherwise, it is non-convex. Self-intersecting 6-polytope are also known as star 6-polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.

• A regular 6-polytope has all identical regular 5-polytope facets. All regular 6-polytope are convex.

Main article: List of regular polytopes#Convex_5

• A semi-regular 6-polytope contains two or more types of regular 4-polytope facets. There is only one such figure, called 2₂₁.

• A uniform 6-polytope has a symmetry group under which all vertices are equivalent, and its facets are uniform 5-polytopes. The edges of a uniform 6-polytope must be equal in length.

Main article: Uniform polypeton

• A prismatic 6-polytope is constructed by a the Cartesian product of two lower-dimensional polytopes. A prismatic 6-polytope is uniform if its factors are uniform. The 6-cube 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 5-space tessellation is the division of five-dimensional Euclidean space into a regular grid of 5-polytope facets. Strictly speaking, tessellations are not 6-polytopes as they do not bound a "6D" volume, but we include them here for the sake of completeness because they are similar in many ways to 6-polytope. A uniform 5-space tessellation is one whose vertices are related by a space group and whose facets are uniform 5-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, 6D, uniform polytopes (polypeta)

External links

• Polytope names
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary
• Glossary for hyperspace, George Olshevsky.