Editing Regular polytope (section)

==History of discovery==<!-- This section is linked from [[Polyhedron]] -->

===Convex polygons and polyhedra===

The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from [[ancient Greece|ancient Greek]] mathematicians. The five [[Platonic solid]]s were known to them. [[Pythagoras]] knew of at least three of them and [[Theaetetus (mathematician)|Theaetetus]] (c. 417 BC – 369 BC) described all five. Later, [[Euclid]] wrote a systematic study of mathematics, publishing it under the title ''[[Euclid's Elements|Elements]]'', which built up a logical theory of geometry and [[number theory]]. His work concluded with mathematical descriptions of the five [[Platonic solid]]s.

:{| class="wikitable"
|-
|colspan=5 align=center|'''[[Platonic solid]]s'''
|-
|align=center|[[Image:Tetrahedron.svg|75px]]
|align=center|[[Image:Hexahedron.svg|75px]]
|align=center|[[Image:Octahedron.svg|75px]]
|align=center|[[Image:POV-Ray-Dodecahedron.svg|75px]]
|align=center|[[Image:Icosahedron.svg|75px]]
|-
|[[Tetrahedron]]||[[Cube]]||[[Octahedron]]||[[Dodecahedron]]||[[Icosahedron]]
|}

===Star polygons and polyhedra===

Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. [[Thomas Bradwardine]] (Bradwardinus) was the first to record a serious study of [[star polygon]]s. Various star polyhedra appear in Renaissance art, but it was not until [[Johannes Kepler]] studied the [[small stellated dodecahedron]] and the [[great stellated dodecahedron]] in 1619 that he realised these two polyhedra were regular. [[Louis Poinsot]] discovered the [[great dodecahedron]] and [[great icosahedron]] in 1809, and [[Augustin Cauchy]] proved the list complete in 1812. These polyhedra are known as collectively as the [[Kepler-Poinsot polyhedra]].

{{main|Regular polyhedron#History}}

{| class="wikitable"
|-
|colspan=4 align=center|'''[[Kepler-Poinsot polyhedron|Kepler-Poinsot polyhedra]]'''
|-
|align=center|[[Image:SmallStellatedDodecahedron.jpg|75px]]
|align=center|[[Image:GreatStellatedDodecahedron.jpg|75px]]
|align=center|[[Image:GreatDodecahedron.jpg|75px]]
|align=center|[[Image:GreatIcosahedron.jpg|75px]]
|-
|[[Small stellated dodecahedron|Small stellated<br>dodecahedron]]||[[Great stellated dodecahedron|Great stellated<br>dodecahedron]]||[[Great dodecahedron]]||[[Great icosahedron]]
|}

===Higher-dimensional polytopes===

[[Image:8-cell-simple.gif|right|thumb|A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. The rotation takes place in the xw plane.]]
It was not until the 19th century that a Swiss mathematician, [[Ludwig Schläfli]], examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in {{Harvtxt|Schläfli|1901}}, six years posthumously, although parts of it were published in {{Harvtxt|Schläfli|1855}} and {{Harvtxt|Schläfli|1858}}. Between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians &mdash; see {{harvtxt|Coxeter|1973|pp=143&ndash;144}} for more details. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by [[Reinhold Hoppe]], one of Schläfli's rediscoverers, in 1882, and first used in English by [[Alicia Boole Stott]] some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).

{{Harvtxt|Coxeter|1973}} is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six [[convex regular 4-polytope|regular convex polytopes in 4 dimensions]]. Five of them can be seen as analogous to the Platonic solids: the [[4-simplex]] (or pentachoron) to the [[tetrahedron]], the [[4-hypercube]] (or 8-cell or [[tesseract]]) to the [[cube]], the [[4-orthoplex]] (or hexadecachoron or [[16-cell]]) to the [[octahedron]], the [[120-cell]] to the [[dodecahedron]], and the [[600-cell]] to the [[icosahedron]].  The sixth, the [[24-cell]], can be seen as a transitional form between the 4-hypercube and 16-cell, analogous to the way that the [[cuboctahedron]] and the [[rhombic dodecahedron]] are transitional forms between the cube and the octahedron.

Also of interest are the star [[regular 4-polytope]]s, partially discovered by Schläfli. By the end of the 19th century, mathematicians such as [[Arthur Cayley]] and [[Ludwig Schläfli]] had developed the theory of regular polytopes in four and higher dimensions, such as the [[tesseract]] and the [[24-cell]].

The latter are difficult (though not impossible) to visualise through a process of [[Four-dimensional space#Dimensional analogy|dimensional analogy]], since they retain the familiar symmetry of their lower-dimensional analogues. The [[tesseract]] contains 8 cubical cells. It consists of two cubes in parallel hyperplanes with corresponding vertices cross-connected in such a way that the 8 cross-edges are equal in length and orthogonal to the 12+12 edges situated on each cube. The corresponding faces of the two cubes are connected to form the remaining 6 cubical faces of the tesseract. The [[24-cell]] can be derived from the tesseract by joining the 8 vertices of each of its cubical faces to an additional vertex to form the four-dimensional analogue of a pyramid. Both figures, as well as other 4-dimensional figures, can be directly visualised and depicted using 4-dimensional stereographs.<ref name="Brisson">{{cite book |editor-first=David W. |editor-last=Brisson |first=David W. |last=Brisson |chapter= Visual Comprehension in n-Dimensions |title=Hypergraphics: Visualizing Complex Relationships In Arts, Science, And Technololgy |chapter-url=https://books.google.com/books?id=zcPADwAAQBAJ&pg=PA109 |date=2019 |publisher=Taylor & Francis |isbn=978-0-429-70681-3 |pages=109–145 |series=AAAS Selected Symposium |volume=24 |orig-year=1978}}</ref>

Harder still to imagine are the more modern [[abstract regular polytope]]s such as the [[57-cell]] or the [[11-cell]]. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives.

In five and more dimensions, there are exactly three finite regular polytopes, which correspond to the tetrahedron, cube and octahedron: these are the [[#Regular simplices|regular simplices]], [[#Measure polytopes (hypercubes)|measure polytopes]] and [[#Cross polytopes (orthoplexes)|cross polytopes]]. Descriptions of these may be found in the [[list of regular polytopes]].

===Elements and symmetry groups===

At the start of the 20th century, the definition of a regular polytope was as follows.
*A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
*A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose [[vertex figure]]s are all congruent and regular.
*And so on, a regular ''n''-polytope is an ''n''-dimensional polytope whose (''n'' &minus; 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.

* An ''n''-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to ''n''&minus;1 dimensions, can be mapped to any other such set by a symmetry of the polytope.

So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, known as a '''[[Flag (geometry)|flag]]''', (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
*A regular polytope is one whose symmetry group is transitive on its flags.

In the 20th century, some important developments were made. The [[symmetry]] [[group (mathematics)|group]]s of the classical regular polytopes were generalised into what are now called [[Coxeter group]]s. Coxeter groups also include the symmetry groups of regular [[tessellation]]s of space or of the plane. For example, the symmetry group of an infinite [[chessboard]] would be the Coxeter group [4,4].

===Apeirotopes &mdash; infinite polytopes===

{{main|Regular skew polyhedron}}

In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures: {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron &mdash; all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges. Nowadays, they are called infinite polyhedra or apeirohedra. The regular tilings of the plane [[Square tiling|{4, 4}]], [[Triangular tiling|{3, 6}]] and [[Hexagonal tiling|{6, 3}]] can also be regarded as infinite polyhedra.

In the 1960s [[Branko Grünbaum]] issued a call to the geometric community to consider more abstract types of regular polytopes that he called ''polystromata''. He developed the theory of polystromata, showing examples of new objects he called [[apeirotope|regular apeirotopes]], that is, regular polytopes with [[infinity|infinitely]] many faces. A simple example of a [[skew apeirogon]] would be a zig-zag. It seems to satisfy the definition of a regular polygon &mdash; all the edges are the same length, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.

===Regular complex polytopes===

{{main|Complex polytope}}

A [[complex number]] has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A complex [[Hilbert space]] has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a '''[[complex polytope]]'''.<ref>Coxeter (1974)</ref>

===Abstract polytopes===

{{main|Abstract polytope}}

[[Image:Hemicube.svg|right|frame|The [[Hemi-cube (geometry)|Hemicube]] is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.]]

Grünbaum also discovered the [[11-cell]], a four-dimensional [[Dual polyhedron|self-dual]] object whose facets are not icosahedra, but are "hemi-icosahedra" &mdash; that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the ''same'' face {{harv|Grünbaum|1976}}. The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.

This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.

The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively curved (elliptic) hyperspace.

A few years after Grünbaum's discovery of the [[11-cell]], [[H. S. M. Coxeter]] independently discovered the same shape. He had earlier discovered a similar polytope, the [[57-cell]] (Coxeter 1982, 1984).

By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called '''[[abstract polytope]]s'''. An abstract polytope is defined as a [[partially ordered set]] (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by ''containment''. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the Platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes.

A geometric polytope is understood to be a ''realization'' of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the corresponding faces of the geometric realisation. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.

The theory has since been further developed, largely by {{harvtxt|McMullen|Schulte|2002}}, but other researchers have also made contributions.

====Regularity of abstract polytopes====
Regularity has a related, though different meaning for [[abstract polytope]]s, since angles and lengths of edges have no meaning.

The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.

Any classical regular polytope has an abstract equivalent which is regular, obtained by taking the set of faces. But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes do not retain information about angles and edge lengths, for example. And a regular abstract polytope may not be realisable as a classical polytope.

''All polygons'' are regular in the abstract world, for example, whereas only those having equal angles and edges of equal length are regular in the classical world.

====Vertex figure of abstract polytopes====

The concept of ''vertex figure'' is also defined differently for an [[abstract polytope]]. The vertex figure of a given abstract ''n''-polytope at a given vertex ''V'' is the set of all abstract faces which contain ''V'', including ''V'' itself.  More formally, it is the abstract section

: ''F''<sub>''n''</sub> / ''V'' = {''F'' | ''V'' ≤ ''F'' ≤ ''F''<sub>''n''</sub>}

where ''F''<sub>''n''</sub> is the maximal face, i.e. the notional ''n''-face which contains all other faces. Note that each ''i''-face, ''i''&nbsp;≥&nbsp;0 of the original polytope becomes an (''i''&nbsp;&minus;&nbsp;1)-face of the vertex figure.

Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures ''may or may not'' be regular itself &ndash; for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons.

The classical vertex figure will, however, be a realisation of the abstract one.