# Template:Polytopes

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1. 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. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets 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, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. 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. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, 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 star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the 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 five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to

2. Uniform polytope – A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons and this is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean, nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension and this approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification, the zeroth rectification is the original form. The th rectification is the dual, an extended Schläfli symbol can be used for representing rectified forms, with a single subscript, k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination, the resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, each higher operation also cuts lower ones too, so a cantellation also truncates vertices. T0,1 or t, Truncation - applied to polygons, a truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges and it can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets, cells are replaced by topologically expanded copies of themselves. There are higher cantellations also, bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc. t0,1,2 or tr, Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of its rectification, a cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves, runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves, There are higher runcinations also, biruncination t1,4, triruncination t2,5, etc. t0,4 or 2r2r, Sterication - applied to Uniform 5-polytopes and higher. It can be seen as birectifying its birectification, Sterication truncates vertices, edges, faces, and cells, replacing each with new facets

3. Coxeter group – In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators

4. Simple Lie group – Simple Lie groups are a class of Lie groups which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative. Many commonly encountered Lie groups are simple or close to being simple, for example. In group theory, a simple Lie group is a locally compact non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0, an equivalent definition of a simple Lie group follows from the Lie correspondence, a connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups and it emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for special examples and configurations in other branches of mathematics. All Lie groups are smooth manifolds, mathematicians often study complex Lie groups, which are Lie groups with a complex structure on the underlying manifold, which is required to be compatible with the group operations. A complex Lie group is called if it is connected as a topological space. Note that the underlying Lie group may not be simple, although it still be semisimple. It is often useful to study slightly more general classes of Lie groups than simple groups, namely semisimple or, more generally, reductive Lie groups. A connected Lie group is called if its Lie algebra is a semisimple lie algebra. It is called if its Lie algebra is a direct sum of simple. Reductive groups occur naturally as symmetries of a number of objects in algebra, geometry. For example, the group G L n of symmetries of a real vector space is reductive. Finite-dimensional representations of simple groups split into direct sums of irreducible representations, simple Lie groups are fully classified. The classification is usually stated in several steps, namely, Classification of simple complex Lie algebras The classification of simple Lie algebras over the numbers by Dynkin diagrams. Classification of centerless Lie groups For every simple Lie algebra g, there is a unique centerless simple Lie group G whose Lie algebra is g, Classification of simple Lie groups One can show that the fundamental group of any Lie group is a discrete commutative group

5. E6 (mathematics) – The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras. This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, the E6 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z. Its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface, the dual representation, which is inequivalent, is also 27-dimensional. In particle physics, E6 plays a role in some grand unified theories, there is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156, the split form, EI, which has maximal compact subgroup Sp/, fundamental group of order 2 and outer automorphism group of order 2. The quasi-split form EII, which has maximal compact subgroup SU × SU/, fundamental group cyclic of order 6, EIII, which has maximal compact subgroup SO × Spin/, fundamental group Z and trivial outer automorphism group. EIV, which has maximal compact subgroup F4, trivial fundamental group cyclic, the EIV form of E6 is the group of collineations of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra, the exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. Over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E6 has exactly one twisted form, known as 2E6, the Dynkin diagram for E6 is given by, which may also be drawn as or. Although they span a space, it is much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space. Two 16-dimensional subalgebras that transform as a Weyl spinor of spin and these have a non-zero last entry. 1 generator which is their chirality generator, and is the sixth Cartan generator, the Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU × SU × SU subalgebra. Other maximal subalgebras which have an importance in physics and can be read off the Dynkin diagram, are the algebras of SO × U, in addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional vector representations. The characters of finite dimensional representations of the real and complex Lie algebras, the fundamental representations have dimensions 27,351,2925,351,27 and 78. The E6 polytope is the hull of the roots of E6. It therefore exists in 6 dimensions, its symmetry group contains the Coxeter group for E6 as an index 2 subgroup, the groups of type E6 over arbitrary fields were introduced by Dickson. The points over a field with q elements of the algebraic group E6, whether of the adjoint or simply connected form

6. E7 (mathematics) – The E7 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z. The dimension of its representation is 56. There is a unique complex Lie algebra of type E7, corresponding to a group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form of E7, the split form, EV, which has maximal compact subgroup SU/, fundamental group cyclic of order 4 and outer automorphism group of order 2. EVI, which has maximal compact subgroup SU·SO/, fundamental group non-cyclic of order 4, EVII, which has maximal compact subgroup SO·E6/, infinite cyclic findamental group and outer automorphism group of order 2. For a complete list of forms of simple Lie algebras. The compact real form of E7 is the group of the 64-dimensional exceptional compact Riemannian symmetric space EVI. This can be seen using a construction known as the magic square, due to Hans Freudenthal. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E7 has no twisted forms, see below. The Dynkin diagram for E7 is given by, even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-dimensional vector space. The roots are all the 8×7 permutations of and all the permutations of Note that the 7-dimensional subspace is the subspace where the sum of all the eight coordinates is zero. The simple roots are We have ordered them so that their corresponding nodes in the Dynkin diagram are ordered left to right with the side node last. Given the E7 Cartan matrix and a Dynkin diagram node ordering of, the Weyl group of E7 is of order 2903040, it is the direct product of the cyclic group of order 2 and the unique simple group of order 1451520. E7 has an SU subalgebra, as is evident by noting that in the 8-dimensional description of the root system, in addition to the 133-dimensional adjoint representation, there is a 56-dimensional vector representation, to be found in the E8 adjoint representation. The characters of finite dimensional representations of the real and complex Lie algebras, there exist non-isomorphic irreducible representation of dimensions 1903725824,16349520330, etc. The fundamental representations are those with dimensions 133,8645,365750,27664,1539,56 and 912, E7 is the automorphism group of the following pair of polynomials in 56 non-commutative variables

7. E8 (mathematics) – The E8 algebra is the largest and most complicated of these exceptional cases. Wilhelm Killing discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them rise to a simple Lie group of dimension 248. Chevalley introduced algebraic groups and Lie algebras of type E8 over other fields, for example, the Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8, therefore, the vectors of the root system are in eight-dimensional Euclidean space, they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the group, has order 21435527 =696729600. There is a Lie algebra Ek for every integer k ≥3, there is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496 and this is simply connected, has maximal compact subgroup the compact form of E8, and has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of forms of simple Lie algebras. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimension 8634368000, the fundamental representations are those with dimensions 3875,6696000,6899079264,146325270,2450240,30380,248 and 147250. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations with the irreducible representations. These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, the most difficult case is the split real form of E8, where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other simple groups have been known for some time. The announcement of the result in March 2007 received extraordinary attention from the media, the representations of the E8 groups over finite fields are given by Deligne–Lusztig theory. One can construct the E8 group as the group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin and it is then possible to check that the Jacobi identity is satisfied

8. F4 (mathematics) – In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups, F4 has rank 4 and dimension 52. The compact form is connected and its outer automorphism group is the trivial group. The compact real form of F4 is the group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen using a construction known as the magic square, due to Hans Freudenthal. There are 3 real forms, a one, a split one. They are the groups of the three real Albert algebras. The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so, in older books and papers, F4 is sometimes denoted by E4. The Dynkin diagram for F4 is and its Weyl/Coxeter group G = W is the symmetry group of the 24-cell, it is a solvable group of order 1152. It has minimal faithful degree μ =24 which is realized by the action on the 24-cell, the F4 lattice is a four-dimensional body-centered cubic lattice. They form a ring called the Hurwitz quaternion ring, the 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin. One choice of roots for F4, is given by the rows of the following matrix. Invariant, F4 is the group of automorphisms of the set of 3 polynomials in 27 variables. Another way of writing these invariants is as Tr, Tr and Tr of the hermitian octonion matrix, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimensions 1053,160056,4313088, etc, the fundamental representations are those with dimensions 52,1274,273,26. The Exceptional Simple Lie Algebras F and E. Proc

9. G2 (mathematics) – In mathematics, G2 is the name of three simple Lie groups, their Lie algebras g 2, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups, G2 has rank 2 and dimension 14. It has two representations, with dimension 7 and 14. The Lie algebra g 2, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23,1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, in the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is related to a ball rolling on another ball. The space of configurations of the ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting. In 1900, Engel discovered that a generic antisymmetric trilinear form on a 7-dimensional complex vector space is preserved by a group isomorphic to the form of G2. In 1908 Cartan mentioned that the group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the real form of G2. In older books and papers, G2 is sometimes denoted by E2, there are 3 simple real Lie algebras associated with this root system, The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an automorphism and is simply connected. The maximal compact subgroup of its associated group is the form of G2. The Lie algebra of the form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, the Lie algebra of the non-compact form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its maximal compact subgroup is SU × SU/. It has a double cover that is simply connected. The Dynkin diagram for G2 is given by and its Cartan matrix is, Although they span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a three-dimensional space

10. H4 (mathematics) – In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators