View source for 5-polytope

{{Short description|5-dimensional geometric object}}
{| align=right class=wikitable
|+ [[Petrie polygon|Graphs]] of three [[List of regular polytopes#Five Dimensions|regular]] and three [[uniform polytope|uniform]] 5-polytopes.
|- align=center valign=top
| [[File:5-simplex t0.svg|150px]]{{br}}[[5-simplex]] (hexateron)
| [[File:5-cube t4.svg|150px]]{{br}}[[5-orthoplex]], {{math|'''2{{sub|11}}'''}}{{br}}(Pentacross)
| [[Image:5-cube_t0.svg|150px]]{{br}}[[5-cube]]{{br}}(Penteract)
|- align=center valign=top
| [[File:5-simplex t04 A4.svg|150px]]{{br}}[[Expanded 5-simplex]]
| [[File:5-cube t3.svg|150px]]{{br}}[[Rectified 5-orthoplex]]
| [[File:5-demicube t0 D5.svg|150px]]{{br}}[[5-demicube]]. {{math|'''1{{sub|21}}'''}}{{br}}(Demipenteract)
|}

In [[geometry]], a '''five-dimensional polytope''' (or '''5-polytope''' or '''polyteron''') is a [[polytope]] in [[five-dimensional space]], bounded by ([[4-polytope]]) [[Facet (geometry)|facets]], pairs of which share a [[Face (geometry)#k-face|polyhedral cell]].

== Definition ==
A 5-polytope is a closed five-dimensional figure with [[Vertex (geometry)|vertices]], [[Edge (geometry)|edges]], [[face (geometry)|faces]], and [[cell (mathematics)|cells]], and [[4-face]]s. A vertex is a [[Point (geometry)|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 [[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 ==
The topology of any given 5-polytope is defined by its [[Betti number]]s and [[torsion coefficient (topology)|torsion coefficient]]s.<ref name="richeson">Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008.</ref>

The value of the [[Euler characteristic]] used to characterise polyhedra does not generalize usefully to higher dimensions, 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.<ref name="richeson"/>

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.<ref name="richeson"/>

== Classification ==

5-polytopes may be classified based on properties like "[[convex set|convexity]]" and "[[symmetry]]".

* A 5-polytope is ''[[Convex polytope|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 polytope]]s, 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 4-polytope]]s.  The faces of a uniform polytope must be [[regular polygon|regular]].

{{Main|Uniform 5-polytope}}

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

{{Main|List_of_regular_polytopes#Convex_4}}

* 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 [[penteract|hypercube]] is prismatic (product of a [[square (geometry)|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 ==

Regular 5-polytopes can be represented by the [[Schläfli symbol]] {{brace|''p'',''q'',''r'',''s''}}, with ''s'' {{brace|''p'',''q'',''r''}} polychoral [[Facet (mathematics)|facets]] around each [[Face (geometry)|face]].

There are exactly three such [[List of regular polytopes#Convex 4|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:

{| class=wikitable
! Name !! [[Schläfli symbol|Schläfli{{br}}symbol]](s)!![[Coxeter diagram|Coxeter{{br}}diagram]](s)!![[Vertex (geometry)|Vertices]]!![[Edge (geometry)|Edges]]!![[Face (geometry)|Faces]]!![[Cell (geometry)|Cells]]!![[4-face]]s!![[Coxeter group|Symmetry]] ([[Symmetry order|order]])
|- align=center
| [[5-simplex]]||{3,3,3,3}||{{CDD|node_1|3|node|3|node|3|node|3|node}}||6||15||20||15||6||A<sub>5</sub>, (120)
|- align=center
|[[5-cube]]||{4,3,3,3}||{{CDD|node_1|4|node|3|node|3|node|3|node}}||32||80||80||40||10||BC<sub>5</sub>, (3820)
|- align=center
| [[5-orthoplex]]||{3,3,3,4}{{br}}{3,3,3<sup>1,1</sup>}||{{CDD|node_1|3|node|3|node|3|node|4|node}}{{br}}{{CDD|node_1|3|node|3|node|split1|nodes}}||10||40||80||80||32||[[Hyperoctahedral group|BC<sub>5</sub>]], (3840){{br}}2×D<sub>5</sub>
|}

== Uniform 5-polytopes ==
{{Main|Uniform 5-polytope}}

For three of the semiregular 5-polytope, their elements are:

{| class=wikitable
!Name!![[Schläfli symbol|Schläfli{{br}}symbol]](s)!![[Coxeter diagram|Coxeter{{br}}diagram]](s)!![[Vertex (geometry)|Vertices]]!![[Edge (geometry)|Edges]]!![[Face (geometry)|Faces]]!![[Cell (geometry)|Cells]]!![[4-face]]s!![[Coxeter group|Symmetry]] ([[Symmetry order|order]])
|- align=center
| [[Expanded 5-simplex]]||t<sub>0,4</sub>{3,3,3,3}||{{CDD|node_1|3|node|3|node|3|node|3|node_1}}||30||120||210||180||162||2×A<sub>5</sub>, (240)
|- align=center
|[[5-demicube]]||{3,3<sup>2,1</sup>}{{br}}h{4,3,3,3}||{{CDD|nodes_10ru|split2|node|3|node|3|node}}{{br}}{{CDD|node_h|4|node|3|node|3|node|3|node}}||16||80||160||120||26||D<sub>5</sub>, (1920){{br}}½BC<sub>5</sub>
|- align=center
| [[Rectified 5-orthoplex]]||t<sub>1</sub>{3,3,3,4}{{br}}t<sub>1</sub>{3,3,3<sup>1,1</sup>}||{{CDD|node|3|node_1|3|node|3|node|4|node}}{{br}}{{CDD|node|3|node_1|3|node|split1|nodes}}||40||240||400||240||42||BC<sub>5</sub>, (3840){{br}}2×D<sub>5</sub>
|}

The ''expanded 5-simplex'' is the [[vertex figure]] of the uniform [[5-simplex honeycomb]], {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}. The [[5-demicube honeycomb]], {{CDD|nodes_10ru|split2|node|3|node|split1|nodes}}, vertex figure is a ''rectified 5-orthoplex'' and  [[Facet (geometry)|facets]] are the ''5-orthoplex'' and ''5-demicube''.

== Pyramids ==

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 is the simplest example with a 4-simplex base.

== See also ==
* [[List of regular polytopes#Five-dimensional regular polytopes and higher]]

== References ==
{{reflist}}

<!-- These need to be changed to inline citations, really... -->
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[Alicia Boole Stott|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
* [[Harold Scott MacDonald Coxeter|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}} [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]
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}}

== External links ==
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
* [http://www.polytope.net/hedrondude/polytera.htm Uniform Polytera], Jonathan Bowers
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary], Garrett Jones

{{Polytopes}}

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[[Category:5-polytopes| ]]