6-polytope explained

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

Definition

A 6-polytope 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 5-polytope. 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.

Characteristics

The topology of any given 6-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, and is zero for all 6-polytopes, 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

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 6-polytope 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.

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 faces of a uniform polytope must be regular.

See main article: Uniform 6-polytope.

A prismatic 6-polytope is constructed by 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.

Regular 6-polytopes

Regular 6-polytopes can be generated from Coxeter groups represented by the Schläfli symbol with t 5-polytope facets around each cell.

There are only three such convex regular 6-polytopes:

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

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

Name	Schläfli symbol	Coxeter diagram	Vertices	Edges	Faces	Cells	4-faces	5-faces!	Symmetry (order)
6-simplex			7	21	35	35	21	7	A₆ (720)
6-orthoplex			12	60	160	240	192	64	B₆ (46080)
6-cube			64	192	240	160	60	12	B₆ (46080)

Uniform 6-polytopes

See main article: Uniform 6-polytope. Here are six simple uniform convex 6-polytopes, including the 6-orthoplex repeated with its alternate construction.

Name	Schläfli symbol(s)	Coxeter diagram(s)	Vertices	Edges	Faces	Cells	4-faces	5-faces!	Symmetry (order)
Expanded 6-simplex	t_0,5		42	210	490	630	434	126	2×A₆ (1440)
6-orthoplex, 3₁₁ (alternate construction)			12	60	160	240	192	64	D₆ (23040)
6-demicube	h		32	240	640	640	252	44	D₆ (23040) ½B₆
Rectified 6-orthoplex	t₁ t₁		60	480	1120	1200	576	76	B₆ (46080) 2×D₆
2₂₁ polytope			27	216	720	1080	648	99	E₆ (51840)
1₂₂ polytope		or	72	720	2160	2160	702	54	2×E₆ (103680)

The expanded 6-simplex is the vertex figure of the uniform 6-simplex honeycomb, . The 6-demicube honeycomb,, vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform 2₂₂ honeycomb,, has 1₂₂ polytope is the vertex figure and 2₂₁ facets.

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (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

External links

Polytope names
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary

Notes and References

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