5-polytope

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Graphs of three regular and three uniform polytopes.
5-simplex t0.svg
5-simplex (hexateron)
5-cube t4.svg
5-orthoplex, 211
(Pentacross)
5-cube t0.svg
5-cube
(Penteract)
5-simplex t04 A4.svg
Expanded 5-simplex
5-cube t3.svg
Rectified 5-orthoplex
5-demicube t0 D5.svg
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[edit]

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[edit]

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[edit]

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} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 6 15 20 15 6 2 A5, (120)
Expanded 5-simplex t0,4{3,3,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 30 120 210 180 162 2 2×A5, (240)
5-demicube {3,32,1}
h{4,3,3,3}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 80 160 120 26 2 D5, (1920)
½BC5
5-cube {4,3,3,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 32 80 80 40 10 2 BC5, (3820)
5-orthoplex {3,3,3,4}
{3,3,31,1}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
10 40 80 80 32 2 BC5, (3840)
2×D5
Rectified 5-orthoplex t1{3,3,3,4}
t1{3,3,31,1}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
40 240 400 240 42 2 BC5, (3840)
2×D5

The expanded 5-simplex is the vertex figure of the uniform 5-simplex honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png. The 5-demicube honeycomb, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube.

Classification[edit]

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[edit]

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[edit]

References[edit]

  • 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)

External links[edit]