Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , where is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension .

Regular polytopes are the generalised analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both mathematicians and non-mathematicians.

Classically, a regular polytope in dimensions may be defined as having regular facets (-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes.

A regular polytope can be represented by a Schläfli symbol of the form with regular facets as and regular vertex figures as

Classification and description

Regular polytopes are classified primarily according to their dimension.

Three classes of regular polytopes exist in every number of dimensions:
*Regular simplex
* Measure polytope (Hypercube)
* Cross polytope (Orthoplex)
Any other regular polytope is said to be exceptional.

In one dimension, the line segment simultaneously serves as the 1-simplex, the 1-hypercube and the 1-orthoplex.

In two dimensions, there are infinitely many regular polygons, namely the regular ''n''-sided polygon for ''n'' ≥ 3. The triangle is the 2-simplex. The square is both the 2-hypercube and the 2-orthoplex. The ''n''-sided polygons for ''n'' ≥ 5 are exceptional.

In three and four dimensions, there are several more exceptional regular polyhedra and 4-polytopes respectively.

In five dimensions and above, the simplex, hypercube and orthoplex are the only regular polytopes. There are no exceptional regular polytopes in these dimensions.

Regular polytopes can be further classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and regular icosahedron. Two distinct regular polytopes with the same symmetry are dual to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries.

The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.

Schläfli symbols

A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

*A convex regular polygon having ''n'' sides is denoted by . So an equilateral triangle is , a square , and so on indefinitely. A regular ''n''-sided star polygon which winds ''m'' times around its centre is denoted by the fractional value , where ''n'' and ''m'' are co-prime, so a regular pentagram is .
*A regular polyhedron having faces with ''p'' faces joining around a vertex is denoted by . The nine regular polyhedra are ; ; ; ; ; ; ; ; and . is the ''vertex figure'' of the polyhedron.
*A regular 4-polytope having cells with ''q'' cells joining around an edge is denoted by . The vertex figure of the 4-polytope is a .
*A regular 5-polytope is an . And so on.

Duality of the regular polytopes

The dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: is self-dual, is dual to , to and so on.

The vertex figure of a regular polytope is the dual of the dual polytope's facet. For example, the vertex figure of is , the dual of which is — a cell of .

The measure and cross polytopes in any dimension are dual to each other.

If the Schläfli symbol is palindromic (i.e. reads the same forwards and backwards), then the polytope is self-dual. The self-dual regular polytopes are:
* All regular polygons - .
* All regular ''n''-simplexes - .
* The regular 24-cell - - in 4 dimensions.
* The great 120-cell - - and grand stellated 120-cell - - in 4 dimensions.
* All regular ''n''-dimensional hypercubic honeycombs - . These may be treated as infinite polytopes.
* Hyperbolic tilings and honeycombs (tilings with p>4 in 2 dimensions; , , , , and in 3 dimensions; in 4 dimensions; and in 5 dimensions).

Regular simplices

These are the regular simplices or simplexes. Their names are, in order of dimension:

:0. Point
:1. Line segment
:2. Equilateral triangle (regular trigon)
:3. Regular tetrahedron (triangular pyramid)
:4. Regular pentachoron ''or'' 4-simplex
:5. Regular hexateron ''or'' 5-simplex
:... An ''n''-simplex has ''n''+1 vertices.

The process of making each simplex can be visualised on a graph: Begin with a point ''A''. Mark point ''B'' at a distance ''r'' from it, and join to form a line segment. Mark point ''C'' in a second, orthogonal, dimension at a distance ''r'' from both, and join to ''A'' and ''B'' to form an equilateral triangle. Mark point ''D'' in a third, orthogonal, dimension a distance ''r'' from all three, and join to form a regular tetrahedron. This process is repeated further using new points to form higher-dimensional simplices.

Measure polytopes (hypercubes)

These are the measure polytopes or hypercubes. Their names are, in order of dimension:

:0. Point
:1. Line segment
:2. Square (regular tetragon)
:3. Cube (regular hexahedron)
:4. Tesseract (regular octachoron) ''or'' 4-cube
:5. Penteract (regular decateron) ''or'' 5-cube
:... An ''n''-cube has ''2ⁿ'' vertices.

The process of making each hypercube can be visualised on a graph: Begin with a point ''A''. Extend a line to point ''B'' at distance ''r'', and join to form a line segment. Extend a second line of length ''r'', orthogonal to ''AB'', from ''B'' to ''C'', and likewise from ''A'' to ''D'', to form a square ''ABCD''. Extend lines of length ''r'' respectively from each corner, orthogonal to both ''AB'' and ''BC'' (i.e. upwards). Mark new points ''E'',''F'',''G'',''H'' to form the cube ''ABCDEFGH''. This process is repeated further using new lines to form higher-dimensional hypercubes.

Cross polytopes (orthoplexes)

These are the cross polytopes or orthoplexes. Their names are, in order of dimensionality:

:0. Point
:1. Line segment
:2. Square (regular tetragon)
:3. Regular octahedron
:4. Regular hexadecachoron ( 16-cell) ''or'' 4-orthoplex
:5. Regular triacontakaiditeron ( pentacross) ''or'' 5-orthoplex
:... An ''n''-orthoplex has ''2n'' vertices.

The process of making each orthoplex can be visualised on a graph: Begin with a point ''O''. Extend a line in opposite directions to points ''A'' and ''B'' a distance ''r'' from ''O'' and 2''r'' apart. Draw a line ''COD'' of length 2''r'', centred on ''O'' and orthogonal to ''AB''. Join the ends to form a square ''ACBD''. Draw a line ''EOF'' of the same length and centered on 'O', orthogonal to ''AB'' and ''CD'' (i.e. upwards and downwards). Join the ends to the square to form a regular octahedron. This process is repeated further using new lines to form higher-dimensional orthoplices.

Classification by Coxeter groups

Regular polytopes can be classified by their isometry group. These are finite Coxeter groups, but not every finite Coxeter group may be realised as the isometry group of a regular polytope. Regular polytopes are in bijection with Coxeter groups with linear Coxeter-Dynkin diagram (without branch point) and an increasing numbering of the nodes. Reversing the numbering gives the dual polytope.

The classification of finite Coxeter groups, which goes back to , therefore implies the classification of regular polytopes:
* Type $A_n$, the symmetric group, gives the regular simplex,
*Type $B_n$, gives the measure polytope and the cross polytope (both can be distinguished by the increasing numbering of the nodes of the Coxeter-Dynkin diagram),
* Exceptional types $I_2(n)$ give the regular polygons (with $n = 3, 4, ...$),
* Exceptional type $H_3$ gives the regular dodecahedron and icosahedron (again the numbering allows to distinguish them),
* Exceptional type $H_4$ gives the 120-cell and the 600-cell,
* Exceptional type $F_4$ gives the 24-cell, which is self-dual.

The bijection between regular polytopes and Coxeter groups with linear Coxeter-Dynkin diagram can be understood as follows. Consider a regular polytope $P$ of dimension $n$ and take its barycentric subdivision. The fundamental domain of the isometry group action on $P$ is given by any simplex $\Delta$ in the barycentric subdivision. The simplex $\Delta$ has $n+1$ vertices which can be numbered from 0 to $n$ by the dimension of the corresponding face of $P$ (the face they are the barycenter of). The isometry group of $P$ is generated by the $n$ reflections around the hyperplanes of $\Delta$ containing the vertex number $n$ (since the barycenter of the whole polytope $P$ is fixed by any isometry). These $n$ hyperplanes can be numbered by the vertex of $\Delta$ they do not contain. The remaining thing to check is that any two hyperplanes with adjacent numbers cannot be orthogonal, whereas hyperplanes with non-adjacent numbers are orthogonal. This can be done using induction (since all facets of $P$ are again regular polytopes). Therefore, the Coxeter-Dynkin diagram of the isometry group of $P$ has $n$ vertices numbered from 0 to $n-1$ such that adjacent numbers are linked by at least one edge and non-adjacent numbers are not linked.

History of discovery

Convex polygons and polyhedra

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

:

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 polygons. 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.

Higher-dimensional polytopes

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 , six years posthumously, although parts of it were published in and . Between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see 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).

is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six 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-polytopes, 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 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.

Harder still to imagine are the more modern abstract regular polytopes 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, measure polytopes and 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 figures are all congruent and regular.
*And so on, a regular ''n''-polytope is an ''n''-dimensional polytope whose (''n'' − 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''−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, (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 groups of the classical regular polytopes were generalised into what are now called Coxeter groups. Coxeter groups also include the symmetry groups of regular tessellations of space or of the plane. For example, the symmetry group of an infinite chessboard would be the Coxeter group ,4

Apeirotopes — infinite polytopes

In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures: , and . They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — 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 , and can also be regarded as infinite polyhedra.

In the 1960s Branko Grünbaum

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