Editing 4-polytope

{{short description|Four-dimensional geometric object with flat sides}}
{|class=wikitable style="float:right; margin-left:8px"
|+ Graphs of the six [[convex regular 4-polytope]]s
|- 
!{3,3,3}
!{3,3,4}
!{4,3,3}
|- valign=top align=center
|[[Image:4-simplex t0.svg|120px]]<BR>[[5-cell]]<BR>Pentatope<BR>4-[[simplex]]
|[[Image:4-cube t3.svg|121px]]<BR>[[16-cell]]<BR>Orthoplex<BR>4-[[orthoplex]]
|[[Image:4-cube t0.svg|120px]]<BR>[[8-cell]]<BR>[[Tesseract]]<BR>4-[[hypercube|cube]]
|- 
!{3,4,3}
!{3,3,5}
!{5,3,3}
|- valign=top align=center
|[[Image:24-cell t0 F4.svg|120px]]<BR>[[24-cell]]<BR>Octaplex
|[[Image:600-cell graph H4.svg|120px]]<BR>[[600-cell]]<BR>Tetraplex
|[[Image:120-cell graph H4.svg|120px]]<BR>[[120-cell]]<BR>Dodecaplex
|}

In [[geometry]], a '''4-polytope''' (sometimes also called a '''polychoron''',<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.1 ''Polytopes and Honeycombs'', p.224</ref> '''polycell''', or '''polyhedroid''') is a [[four-dimensional]] [[polytope]].<ref>{{Cite book | last = Vialar | first = T. | title = Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance | publisher = Springer | year = 2009 | page = 674
  | url = https://books.google.com/books?id=uf20taaf-VgC&q=polychoron&pg=PA674 | isbn = 978-3-540-85977-2}}</ref><ref>{{Cite book
  | last = Capecchi | first = V. |author2=Contucci, P. |author3=Buscema, M. |author4=D'Amore, B.  | title = Applications of Mathematics in Models, Artificial Neural Networks and Arts
  | publisher = Springer | year = 2010 | page = 598 | url = https://books.google.com/books?id=oNy5MxGXLEwC&q=polychoron&pg=PA598 | doi = 10.1007/978-90-481-8581-8
  | isbn = 978-90-481-8580-1}}</ref> It is a connected and closed figure, composed of lower-dimensional polytopal elements: [[Vertex (geometry)|vertices]], [[Edge (geometry)|edges]], [[Face (geometry)|faces]] ([[polygon]]s), and [[Cell (mathematics)|cells]] ([[Polyhedron|polyhedra]]). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician [[Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|p=141|loc=§7-x. Historical remarks}} 

The two-dimensional analogue of a 4-polytope is a [[polygon]], and the three-dimensional analogue is a [[polyhedron]].

Topologically 4-polytopes are closely related to the [[Convex uniform honeycomb|uniform honeycombs]], such as the [[cubic honeycomb]], which tessellate 3-space; similarly the 3D [[cube]] is related to the infinite 2D [[square tiling]]. Convex 4-polytopes can be ''cut and unfolded'' as [[polyhedral net|nets]] in 3-space.

==Definition==
A 4-polytope is a closed [[Four-dimensional space|four-dimension]]al figure. It comprises [[vertex (geometry)|vertices]] (corner points), [[edge (geometry)|edges]], [[face (geometry)|faces]] and [[cell (mathematics)|cells]]. A cell is the three-dimensional analogue of a face, and is therefore a [[polyhedron]]. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.

==Geometry==
The convex [[regular 4-polytopes]] are the four-dimensional analogues of the [[Platonic solids]]. The most familiar 4-polytope is the [[tesseract]] or hypercube, the 4D analogue of the cube.

The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=: [An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.]}} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[Regular 4-polytope#As configurations|configuration matrices]] or simply the number of vertices) follows the same ordering.

{{Regular convex 4-polytopes}}

==Visualisation==
{| class=wikitable align=right
|+ Example presentations of a [[24-cell]]
!colspan=2|Sectioning
![[Net (polytope)|Net]]
|- align=center
|colspan=2|[[File:24cell section anim.gif|200px]]
|[[File:Polychoron 24-cell net.png|150px]]
|-
!colspan=3|Projections
|-
![[Schlegel diagram|Schlegel]]
!2D orthogonal
!3D orthogonal
|- align=center
|[[File:Schlegel wireframe 24-cell.png|100px]]
|[[File:24-cell t0 F4.svg|100px]]
|[[File:Orthogonal projection envelopes 24-cell.png|150px]]
|}

4-polytopes cannot be seen in three-dimensional space due to their extra dimension. Several techniques are used to help visualise them.

;Orthogonal projection
[[Orthogonal projection]]s can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible [[projective envelope]]s.
;Perspective projection
Just as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or even onto a flat sheet. One common projection is a [[Schlegel diagram]] which uses [[stereographic projection]] of points on the surface of a [[3-sphere]] into three dimensions, connected by straight edges, faces, and cells drawn in 3-space.

;Sectioning
Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce a smooth animation of these cross sections.

;Nets
A [[Net (polytope)|net]] of a 4-polytope is composed of polyhedral [[Cell (geometry)|cells]] that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a [[net (polyhedron)|net of a polyhedron]] are connected by their edges and all occupy the same plane.

== Topological characteristics ==
[[Image:Hypercube.svg|150px|thumb|The [[tesseract]] as a [[Schlegel diagram]]]]

The topology of any given 4-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, and is zero for all 4-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.<ref name="richeson"/>

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.<ref name="richeson"/>

==Classification==

===Criteria===
Like all polytopes, 4-polytopes may be classified based on properties like "[[convex set|convexity]]" and "[[symmetry]]".

*A 4-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 4-polytope is contained in the 4-polytope or its interior; otherwise, it is ''non-convex''.  Self-intersecting 4-polytopes are also known as [[star 4-polytope]]s, from analogy with the star-like shapes of the non-convex [[star polygon]]s and [[Kepler–Poinsot polyhedra]].
* A 4-polytope is ''[[regular polytope|regular]]'' if it is [[transitive group action|transitive]] on its [[Flag (geometry)|flags]]. This means that its cells are all [[congruence (geometry)|congruent]] [[regular polyhedra]], and similarly its [[vertex figures]] are congruent and of another kind of regular polyhedron.
*A convex 4-polytope is ''[[semiregular polytope|semi-regular]]'' if it has a [[symmetry group]] under which all vertices are equivalent ([[vertex-transitive]]) and its cells are [[regular polyhedron|regular polyhedra]]. The cells may be of two or more kinds, provided that they have the same kind of face. There are only 3 cases identified by [[Thorold Gosset]] in 1900: the [[rectified 5-cell]], [[rectified 600-cell]], and [[snub 24-cell]].
*A 4-polytope is ''[[uniform polytope|uniform]]'' if it has a [[symmetry group]] under which all vertices are equivalent, and its cells are [[uniform polyhedron|uniform polyhedra]]. The faces of a uniform 4-polytope must be [[regular polygon|regular]].
* A 4-polytope is ''[[scaliform polytope|scaliform]]'' if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex [[Johnson solid]]s.
*A regular 4-polytope which is also [[convex polytope|convex]] is said to be a [[convex regular 4-polytope]].
*A 4-polytope is ''prismatic'' if it is the [[Cartesian product]] of two or more lower-dimensional polytopes. A prismatic 4-polytope is uniform if its factors are uniform. The [[tesseract|hypercube]] is prismatic (product of two [[square (geometry)|square]]s, or of a [[cube]] and [[line segment]]), but is considered separately because it has symmetries other than those inherited from its factors.
*A ''[[tessellation|tiling]] or [[honeycomb (geometry)|honeycomb]] of 3-space'' is the division of three-dimensional [[Euclidean space]] into a repetitive [[Grid (spatial index)|grid]] of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A ''uniform tiling of 3-space'' is one whose vertices are congruent and related by a [[space group]] and whose cells are [[uniform polyhedron|uniform polyhedra]].

=== Classes ===

The following lists the various categories of 4-polytopes classified according to the criteria above:
[[File:Schlegel half-solid truncated 120-cell.png|150px|thumb|The [[truncated 120-cell]] is one of 47 convex non-prismatic uniform 4-polytopes]]
'''[[Uniform 4-polytope]]''' ([[vertex-transitive]]):
* '''Convex uniform 4-polytopes''' (64, plus two infinite families)
** 47 non-prismatic [[uniform 4-polytope#Convex uniform 4-polytope|convex uniform 4-polytope]] including:
*** 6 [[Convex regular 4-polytope]]
** [[uniform 4-polytope#Prismatic uniform 4-polytope|Prismatic uniform 4-polytopes]]:
*** {} × {p,q} : 18 [[Uniform 4-polytope#Polyhedral hyperprisms|polyhedral hyperprisms]] (including cubic hyperprism, the regular [[hypercube]])
*** Prisms built on antiprisms (infinite family)
*** {p} × {q} : [[duoprism]]s (infinite family)
* '''Non-convex uniform 4-polytopes''' (10 + unknown)[[File:Ortho solid 016-uniform polychoron p33-t0.png|150px|thumb|The [[great grand stellated 120-cell]] is the largest of 10 regular star 4-polytopes, having 600 vertices.]]
** 10 (regular) [[Schläfli-Hess polytope]]s
** 57 hyperprisms built on [[Nonconvex uniform polyhedron|nonconvex uniform polyhedra]]
** Unknown total number of nonconvex uniform 4-polytopes: [[Norman Johnson (mathematician)|Norman Johnson]] and other collaborators have identified 2191 forms (convex and star, excluding the infinite families), all constructed by [[vertex figures]] by [[Stella (software)|Stella4D software]].<ref>[https://www.mit.edu/~hlb/Associahedron/program.pdf Uniform Polychora], Norman W. Johnson (Wheaton College), 1845 cases in 2005</ref>

'''Other convex 4-polytopes''':
* [[Polyhedral pyramid]]
* [[Polyhedral bipyramid]]
* [[Polyhedral prism]]
<!--* [[Polyhedral antiprism]]-->
[[File:Cubic honeycomb.png|150px|thumb|The regular [[cubic honeycomb]] is the only infinite regular 4-polytope in Euclidean 3-dimensional space.]]
'''Infinite uniform 4-polytopes of [[Euclidean space|Euclidean 3-space]]''' (uniform tessellations of convex uniform cells)
* 28 [[convex uniform honeycomb]]s: uniform convex polyhedral tessellations, including:
** 1 regular tessellation, [[cubic honeycomb]]: {4,3,4}

'''Infinite uniform 4-polytopes of [[Hyperbolic space|hyperbolic 3-space]]''' (uniform tessellations of convex uniform cells)
* 76 Wythoffian [[convex uniform honeycombs in hyperbolic space]], including:
** [[List of regular polytopes#Tessellations of hyperbolic 3-space|4 regular tessellation of compact hyperbolic 3-space]]: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}

'''Dual [[uniform 4-polytope]]''' ([[cell-transitive]]):
* 41 unique dual convex uniform 4-polytopes
* 17 unique dual convex uniform polyhedral prisms
* infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
* 27 unique convex dual uniform honeycombs, including:
** [[Rhombic dodecahedral honeycomb]]
** [[Disphenoid tetrahedral honeycomb]]

'''Others:'''
* [[Weaire–Phelan structure]] periodic space-filling honeycomb with irregular cells
[[File:Hemi-icosahedron coloured.svg|150px|thumb|The [[11-cell]] is an abstract regular 4-polytope, existing in the [[real projective plane]], it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.]]
'''[[Abstract polytope|Abstract regular 4-polytopes]]''':
* [[11-cell]]
* [[57-cell]]

These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.

==See also==

*[[Regular 4-polytope]]
*[[3-sphere]] – analogue of a sphere in 4-dimensional space. This is not a 4-polytope, since it is not bounded by polyhedral cells.
*The [[duocylinder]] is a figure in 4-dimensional space related to the [[duoprism]]s. It is also not a 4-polytope because its bounding volumes are not polyhedral.

== References ==

=== Notes ===
{{Reflist}}
{{notelist}}

=== Bibliography ===
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
** H.S.M. Coxeter, M.S. Longuet-Higgins and [[J.C.P. Miller]]: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
** '''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]
* [[John Horton Conway|J.H. Conway]] and [[Michael Guy (computer scientist)|M.J.T. Guy]]: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation  [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }}

==External links==
{{Commons category}}
*{{Mathworld | urlname=Polychoron | title=Polychoron }}
*{{Mathworld | urlname=PolyhedralFormula | title=Polyhedral formula }}
*{{Mathworld | urlname=RegularPolychoron | title=Regular polychoron Euler characteristics}}
*[http://www.polytope.net/hedrondude/polychora.htm Uniform Polychora], Jonathan Bowers
*[https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Uniform polychoron Viewer - Java3D Applet with sources]
* R. Klitzing, [http://www.bendwavy.org/klitzing/dimensions/polychora-neu.htm polychora]

{{Polytopes}}

[[Category:Four-dimensional geometry]]
[[Category:Algebraic topology]]
[[Category:4-polytopes| ]]