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Coxeter-Dynkin Diagram
In geometry, a Coxeter– Dynkin diagram
Dynkin diagram
(or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6
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Geometry
Geometry
Geometry
(from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes
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G2 (mathematics)
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 displaystyle mathfrak 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
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Greatest Common Divisor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers
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Group Order
In group theory, a branch of mathematics, the term order is used in two unrelated senses:The order of a group is its cardinality, i.e., the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. The ordering relation of a partially or totally ordered group.This article is about the first sense of order. The order of a group G is denoted by ord(G) or G and the order of an element a is denoted by ord(a) or a.Contents1 Example 2 Order and structure 3 Counting by order of elements 4 In relation to homomorphisms 5 Class equation 6 Open questions 7 See also 8 ReferencesExample[edit] Example
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Affine Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). The multiple edges are, within certain constraints, directed. The main interest in Dynkin diagrams are as a means to classify semisimple Lie algebras over algebraically closed fields. This gives rise to Weyl groups, i.e. to many (although not all) finite reflection groups. Dynkin diagrams may also arise in other contexts. The term "Dynkin diagram" can be ambiguous
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Fundamental Domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space
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Hyperbolic Plane
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry
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Simple Lie Group
In group theory, a simple Lie group
Lie group
is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple Lie algebra
Lie algebra
is a non-abelian Lie algebra
Lie algebra
whose only ideals are 0 and itself (or equivalently, a Lie algebra
Lie algebra
of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups
Lie groups
are a class of Lie groups
Lie groups
which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups
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Exceptional Lie Group
In group theory, a simple Lie group
Lie group
is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple Lie algebra
Lie algebra
is a non-abelian Lie algebra
Lie algebra
whose only ideals are 0 and itself (or equivalently, a Lie algebra
Lie algebra
of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups
Lie groups
are a class of Lie groups
Lie groups
which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups
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En (Lie Algebra)
In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram
Dynkin diagram
is a bifurcating graph with three branches of length 1,2, and k, with k=n-4. In some older books and papers, E2 and E4 are used as names for G2 and F4.Contents1 Finite-dimensional Lie algebras 2 Infinite-dimensional Lie algebras 3 Root lattice 4 E7½ 5 See also 6 References 7 Further readingFinite-dimensional Lie algebras[edit] The En group is similar to the An group, except the nth node is connected to the 3rd node. So the Cartan matrix
Cartan matrix
appears similar, -1 above and below the diagonal, except for the last row and column, have -1 in the third row and column
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F4 (mathematics)
In mathematics, F4 is the name of a Lie group
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 simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional. The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2
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Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle.[1][2][3] A rectangle with vertices ABCD would be denoted as  ABCD. The word rectangle comes from the Latin
Latin
rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals.[4] It is a special case of an antiparallelogram, and its angles are not right angles
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Cartan Matrix
In mathematics, the term Cartan matrix
Cartan matrix
has three meanings. All of these are named after the French mathematician Élie Cartan
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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 (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions
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Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.[1] In a polygon, an edge is a line segment on the boundary,[2] and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet.[3] A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.Contents1 Relation to edges in graphs 2 Number of edges in a polyhedron 3 Incidences with other faces 4 Alternative terminology 5 See also 6 References 7 External linksRelation to edges in graphs[edit] In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment
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