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In five-dimensional geometry, a five-dimensional polytope or 5-polytope
5-polytope
is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

Contents

1 Definition 2 Characteristics 3 Classification 4 Regular 5-polytopes 5 Uniform 5-polytopes 6 Pyramids 7 See also 8 References 9 External links

Definition[edit] A 5-polytope
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
4-face
is a 4-polytope. Furthermore, the following requirements must be met:

Each cell must join exactly two 4-faces. Adjacent 4-faces are not in the same four-dimensional hyperplane. The figure is not a compound of other figures which meet the requirements.

Characteristics[edit] The topology of any given 5-polytope
5-polytope
is defined by its Betti numbers and torsion coefficients.[1] The value of the Euler characteristic
Euler characteristic
used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic
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[edit] 5-polytopes may be classified based on properties like "convexity" and "symmetry".

A 5-polytope
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
5-polytope
is contained in the 5-polytope
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
5-polytope
has a symmetry group under which all vertices are equivalent, and its facets are uniform 4-polytopes. The faces of a uniform polytope must be regular.

Main article: Uniform 5-polytope

A semi-regular 5-polytope
5-polytope
contains two or more types of regular 4-polytope
4-polytope
facets. There is only one such figure, called a demipenteract. A regular 5-polytope
5-polytope
has all identical regular 4-polytope
4-polytope
facets. All regular 5-polytopes are convex.

Main article: List_of_regular_polytopes § Convex_4

A prismatic 5-polytope
5-polytope
is constructed by a Cartesian product
Cartesian product
of two lower-dimensional polytopes. A prismatic 5-polytope
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 polytopes 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 polytopes. A uniform 4-space tessellation is one whose vertices are related by a space group and whose facets are uniform 4-polytopes.

Regular 5-polytopes[edit] Regular 5-polytopes can be represented by the Schläfli symbol
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 - 5-simplex 4,3,3,3 - 5-cube 3,3,3,4 - 5-orthoplex

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 Symmetry
Symmetry
(order)

5-simplex 3,3,3,3

6 15 20 15 6 A5, (120)

5-cube 4,3,3,3

32 80 80 40 10 BC5, (3820)

5-orthoplex 3,3,3,4 3,3,31,1

10 40 80 80 32 BC5, (3840) 2×D5

Uniform 5-polytopes[edit] Main article: Uniform 5-polytope For three of the semiregular 5-polytope, their elements are:

Name Schläfli symbol(s) Coxeter diagram(s) Vertices Edges Faces Cells 4-faces Symmetry
Symmetry
(order)

Expanded 5-simplex t0,4 3,3,3,3

30 120 210 180 162 2×A5, (240)

5-demicube 3,32,1 h 4,3,3,3

16 80 160 120 26 D5, (1920) ½BC5

Rectified 5-orthoplex t1 3,3,3,4 t1 3,3,31,1

40 240 400 240 42 BC5, (3840) 2×D5

The expanded 5-simplex
5-simplex
is the vertex figure of the uniform 5-simplex honeycomb, . The 5-demicube
5-demicube
honeycomb, , vertex figure is a rectified 5-orthoplex
5-orthoplex
and facets are the 5-orthoplex
5-orthoplex
and 5-demicube. Pyramids[edit] Pyramidal 5-polytopes, or 5-pyramids, can be generated by a 4-polytope base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex
5-simplex
is the simplest example with a 4-simplex base. See also[edit]

List of regular polytopes#Five-dimensional regular polytopes and higher

References[edit]

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

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 Klitzing, Richard. "5D uniform polytopes (polytera)". 

External links[edit]

Polytopes of Various Dimensions, Jonathan Bowers Uniform Polytera, Jonathan Bowers Multi-dimensional Glossary, Garrett Jones

v t e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn

Regular polygon Triangle Square p-gon Hexagon Pentagon

Uniform polyhedron Tetrahedron Octahedron
Octahedron
• Cube Demicube

Dodecahedron • Icosahedron

Uniform 4-polytope 5-cell 16-cell
16-cell
• Tesseract Demitesseract 24-cell 120-cell
120-cell
• 600-cell

Uniform 5-polytope 5-simplex 5-orthoplex
5-orthoplex
• 5-cube 5-demicube

Uniform 6-polytope 6-simplex 6-orthoplex
6-orthoplex
• 6-cube 6-demicube 122 • 221

Uniform 7-polytope 7-simplex 7-orthoplex
7-orthoplex
• 7-cube 7-demicube 132 • 231 • 321

Uniform 8-polytope 8-simplex 8-orthoplex
8-orthoplex
• 8-cube 8-demicube 142 • 241 • 421

Uniform 9-polytope 9-simplex 9-orthoplex
9-orthoplex
• 9-cube 9-demicube

Uniform 10-polytope 10-simplex 10-orthoplex
10-orthoplex
• 10-cube 10-demicube

Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope

Topics: Polytope
Polytope
families • Regular polytope
Regular polytope
• List of regular polyt

.