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Coxeter-Dynkin Diagram
In geometry , a COXETER–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. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras
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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 arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages . By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat . Since then, and into modern times, geometry has expanded into non- Euclidean geometry and manifolds , describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry
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Graph (discrete Mathematics)
In mathematics , and more specifically in graph theory , a GRAPH is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called an arc or line). Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics . The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. In contrast, if any edge from a person A to a person B corresponds to A's admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges. Graphs are the basic subject studied by graph theory . The word "graph" was first used in this sense by J. J. Sylvester in 1878
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Mirror
A MIRROR is an object that reflects light in such a way that, for incident light in some range of wavelengths, the reflected light preserves many or most of the detailed physical characteristics of the original light. This is different from other light-reflecting objects that do not preserve much of the original wave signal other than color and diffuse reflected light. The most familiar type of mirror is the plane mirror , which has a flat surface. Curved mirrors are also used, to produce magnified or diminished images or focus light or simply distort the reflected image. Mirrors are commonly used for personal grooming or admiring oneself (where they are also called LOOKING-GLASSES), decoration, and architecture. Mirrors are also used in scientific apparatus such as telescopes and lasers , cameras, and industrial machinery. Most mirrors are designed for visible light ; however, mirrors designed for other wavelengths of electromagnetic radiation are also used
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Hyperplane
In geometry a HYPERPLANE is a subspace of one dimension less than its ambient space . If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes , while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines . This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces . But a hyperplane of an n-dimensional projective space does not have this property. CONTENTS * 1 Technical description * 2 Special
Special
types of hyperplanes * 2.1 Affine hyperplanes * 2.2 Vector hyperplanes * 2.3 Projective hyperplanes * 3 Dihedral angles * 3.1 Support hyperplanes * 4 See also * 5 References * 6 External links TECHNICAL DESCRIPTIONIn geometry , a HYPERPLANE of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V
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Kaleidoscope
A KALEIDOSCOPE is an optical instrument with two or more reflecting surfaces inclined to each other in an angle, so that one or more (parts of) objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection . The reflectors (or mirrors ) are usually enclosed in a tube, often containing on one end a cell with loose, colored pieces of glass or other transparent (and/or opaque) materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing viewed pattern. CONTENTS * 1 Etymology * 2 Principles * 3 History * 4 Variations * 4.1 General variables * 4.2 Different versions suggested by Brewster * 4.3 Later variations * 5 Publications * 6 Applications * 7 See also * 8 References * 9 External links ETYMOLOGYCoined in 1817 by Scottish inventor David Brewster , "kaleidoscope" is derived from the Ancient Greek καλός (kalos), "beautiful, beauty", εἶδος (eidos), "that which is seen: form, shape" and σκοπέω (skopeō), "to look to, to examine", hence "observation of beautiful forms." PRINCIPLES The basic configuration of reflecting surfaces in the kaleidoscope, as illustrated in the 1817 patent. Fig. 2 and Fig. 3 show alternative shapes of the reflectors
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Facet (mathematics)
In geometry , a FACET is a feature of a polyhedron , polytope , or related geometric structure, generally of dimension one less than the structure itself. * In three-dimensional geometry a FACET of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . To FACET a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes . * In polyhedral combinatorics and in the general theory of polytopes , a FACET of a polytope of dimension n is a face that has dimension n − 1. Facets may also be called (n − 1)-faces. In three-dimensional geometry, they are often called "faces" without qualification. * A FACET of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For simplicial polytopes this coincides with the meaning from polyhedral combinatorics.REFERENCES * ^ Bridge, N.J. Facetting the dodecahedron, Acta crystallographica A30 (1974), pp. 548–552. * ^ Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253–261. * ^ Coxeter, H. S. M. (1973), Regular Polytopes, Dover, p. 95 . * ^ Matoušek, Jiří (2002), Lectures in Discrete Geometry, Graduate Texts in Mathematics , 212, Springer, 5.3 Faces of a Convex Polytope, p. 86 . * ^ De Loera, Jesús A
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Dihedral Angle
A DIHEDRAL ANGLE is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge . In higher dimension, a dihedral angle represents the angle between two hyperplanes . CONTENTS * 1 Definitions * 2 Dihedral angles in stereochemistry * 3 Dihedral angles of proteins * 3.1 Converting from dihedral angles to Cartesian coordinates in chains * 4 Calculation of a dihedral angle * 5 Dihedral angles in polyhedra * 6 See also * 7 References * 8 External links DEFINITIONSA dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection. A torsion angle is a particular example of a dihedral angle, used in stereochemistry to define the geometric relation of two parts of a molecule joined by a chemical bond . DIHEDRAL ANGLES IN STEREOCHEMISTRY See also: Alkane stereochemistry and Conformational isomerism _ Configuration names syn_ _n-_ Butane Newman projection _syn_ _n-_ Butane sawhorse projection Free energy diagram of butane as a function of dihedral angle. In stereochemistry every set of three (not co-linear) atoms of a molecule defines a plane
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Ridge (geometry)
In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes , the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). CONTENTS* 1 Polygonal face * 1.1 Number of polygonal faces of a polyhedron * 2 k-face * 2.1 Cell or 3-face
3-face
* 2.2 Facet or (n-1)-face * 2.3 Ridge or (n-2)-face * 2.4 Peak or (n-3)-face * 3 See also * 4 References * 5 External links POLYGONAL FACEIn elementary geometry, a FACE is a polygon on the boundary of a polyhedron . Other names for a polygonal face include SIDE of a polyhedron, and TILE of a Euclidean plane tessellation . For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope
4-polytope
. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4-POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex
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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 (or kaleidoscopic mirrors ). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 ). Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes , and the Weyl groups of simple Lie algebras . Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane , and the Weyl groups of infinite-dimensional Kac–Moody algebras . Standard references include (Humphreys 1992 ) and (Davis 2007 )
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Dynkin Diagram
In the mathematical field of Lie theory , a DYNKIN DIAGRAM, named for Eugene Dynkin
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. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups; the B n {displaystyle B_{n}} and C n {displaystyle C_{n}} directed diagrams yield the same undirected diagram, correspondingly named B C n . {displaystyle BC_{n}.} In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named
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Directed Graph
In mathematics , and more specifically in graph theory , a DIRECTED GRAPH (or DIGRAPH) is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them. CONTENTS * 1 Definition * 2 Types of directed graphs * 2.1 Subclasses * 2.2 Digraphs with supplementary properties * 3 Basic terminology * 4 Indegree and outdegree * 5 Degree sequence * 6 Directed graph
Directed graph
connectivity * 7 See also * 8 Notes * 9 References DEFINITIONIn formal terms, a directed graph is an ordered pair G = (V, A) where * V is a set whose elements are called vertices, nodes, or points; * A is a set of ordered pairs of vertices, called arrows, directed edges (sometimes simply edges with the corresponding set named E instead of A), directed arcs, or directed lines.It differs from an ordinary or undirected graph , in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, arcs, or lines
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Undirected Graph
In mathematics , and more specifically in graph theory , a GRAPH is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called an arc or line). Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics . The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. In contrast, if any edge from a person A to a person B corresponds to A's admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges. Graphs are the basic subject studied by graph theory . The word "graph" was first used in this sense by J. J. Sylvester in 1878
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Crystallographic Restriction Theorem
The CRYSTALLOGRAPHIC RESTRICTION THEOREM in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman . Crystals are modeled as discrete lattices , generated by a list of independent finite translations (Coxeter 1989 ). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry). The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups. CONTENTS* 1 Dimensions 2 and 3 * 1.1 Lattice proof * 1.2 Trigonometry proof * 1.3 Short trigonometry proof * 1.4 Matrix proof * 2 Higher dimensions * 3 Formulation in terms of isometries * 4 See also * 5 Notes * 6 References * 7 External links DIMENSIONS 2 AND 3The special cases of 2D (wallpaper groups ) and 3D (space groups ) are most heavily used in applications, and they can be treated together
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Root Systems
In mathematics , a ROOT SYSTEM is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras , especially the classification and representations theory of semisimple Lie algebras . Since Lie groups (and some analogues such as algebraic groups ) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams , occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory ). Finally, root systems are important for their own sake, as in spectral graph theory
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Semisimple Lie Algebra
In mathematics , a Lie algebra
Lie algebra
is SEMISIMPLE if it is a direct sum of simple Lie algebras , i.e., non-abelian Lie algebras g {displaystyle {mathfrak {g}}} whose only ideals are {0} and g {displaystyle {mathfrak {g}}} itself. Throughout the article, unless otherwise stated, g {displaystyle {mathfrak {g}}} is a non-zero finite-dimensional Lie algebra over a field of characteristic 0