5-polytope
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:
- Each cell must join exactly two hypercells.
- Adjacent hypercells are not in the same four-dimensional hyperplane.
- 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:
- {3,3,3,3} - Hexateron (5-simplex)
- {4,3,3,3} - Penteract (5-hypercube)
- {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.
- A regular 5-polytope has all identical regular polychoron facets. All regular polytera are convex.
- 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
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