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
Description
Regular polytopes are classified primarily according to their dimension. 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 -sided polygon for . The triangle is the 2-simplex. The square is both the 2-hypercube and the 2-orthoplex. The -sided polygons for 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 objects. 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 sides is denoted by . So, an equilateral triangle is , a square is , and so on, indefinitely. A regular -sided star polygon which winds times around its centre is denoted by the fractional value {n/m}, where n and m are co-prime, so a regular pentagram is {5/2}. A regular polyhedron having faces with faces joining around a vertex is denoted by . The nine regular polyhedra are the five Platonic solids—tetrahedron , octahedron , cube , icosahedron , dodecahedron — and the four star polyhedra—great icosahedron , great stellated dodecahedron , great dodecahedron , small stellated dodecahedron . This notation can be used in regular tilings, as in triangular tiling , hexagonal tiling , and square tiling . is the vertex figure of the polyhedron. A regular 4-polytope having cells {n, p} with q cells joining around an edge is denoted by {n, p, q}. The vertex figure of the 4-polytope is a {p, q}. A regular 5-polytope is denoted by {n, p, q, r}, 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: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} 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 {3, 3, 4} is {3, 4}, the dual of which is {4, 3} — a cell of {4, 3, 3}.
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 - {a}. All regular n-simplexes - {3,3,...,3}. The regular 24-cell - {3,4,3} - in 4 dimensions. The great 120-cell - {5,5/2,5} - and grand stellated 120-cell - {5/2,5,5/2} - in 4 dimensions. All regular n-dimensional hypercubic honeycombs - {4,3,...,3,4}. These may be treated as infinite polytopes. Hyperbolic tilings and honeycombs (tilings {p,p} with p>4 in 2 dimensions; {4,4,4}, {5,3,5}, {3,5,3}, {6,3,6}, and {3,6,3} in 3 dimensions; {5,3,3,5} in 4 dimensions; and {3,3,4,3,3} in 5 dimensions).
Classifications
Regular simplices
The simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex: point a 1-dimensional simplex: line segment, obtained by connecting another point at a distance a 2-dimensional simplex is a triangle, obtained by connecting two points to another point a 3-dimensional simplex is a tetrahedron, obtained by connecting three points to another point, again, and a 4-dimensional simplex is a 5-cell. This process is repeated further using new points to form higher-dimensional simplices.
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.
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 2r apart. Draw a line COD of length 2r, 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 , the symmetric group, gives the regular simplex, Type , 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 give the regular polygons (with ), Exceptional type gives the regular dodecahedron and icosahedron (again the numbering allows to distinguish them), Exceptional type gives the 120-cell and the 600-cell, Exceptional type 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 of dimension and take its barycentric subdivision. The fundamental domain of the isometry group action on is given by any simplex in the barycentric subdivision. The simplex has vertices which can be numbered from 0 to by the dimension of the corresponding face of (the face they are the barycenter of). The isometry group of is generated by the reflections around the hyperplanes of containing the vertex number (since the barycenter of the whole polytope is fixed by any isometry). These hyperplanes can be numbered by the vertex of 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 are again regular polytopes). Therefore, the Coxeter-Dynkin diagram of the isometry group of has vertices numbered from 0 to 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 inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.
Star polygons and polyhedra
The understanding of the solids 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 collectively known 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. 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.