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Vertex Figure
In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance
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Convex Polytope
A CONVEX POLYTOPE is a special case of a polytope , having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "CONVEX POLYHEDRON" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set , while others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n-1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming . A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum
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Coxeter 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
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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Eric W. Weisstein
ERIC WOLFGANG WEISSTEIN (born March 18, 1969) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science ( ScienceWorld ). He is the author of the CRC Concise Encyclopedia of Mathematics . He currently works for Wolfram Research , Inc. CONTENTS * 1 Education * 2 Career * 2.1 Academic research * 2.2 MathWorld, ScienceWorld and Wolfram Research * 2.3 Further scientific activities * 3 Footnotes * 4 References * 5 External links EDUCATION Weisstein holds a Ph.D. in planetary astronomy which he obtained from the California Institute of Technology 's (Caltech) Division of Geological and Planetary Sciences in 1996 as well as an M.S. in planetary astronomy in 1993 also from Caltech. Weisstein graduated Cum Laude from Cornell University with a B.A. in physics and a minor in astronomy in 1990
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Honeycomb (geometry)
In geometry , a HONEYCOMB is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces , such as hyperbolic honeycombs . Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles , as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell
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Facet (geometry)
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
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Polygon
In elementary geometry , a POLYGON (/ˈpɒlɪɡɒn/ ) is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An N-GON is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other self-intersecting polygons
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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
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Graph Theory
In mathematics , GRAPH THEORY is the study of graphs , which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices , nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics . Refer to the glossary of graph theory for basic definitions in graph theory
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place
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Harold Scott MacDonald Coxeter
HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London
London
but spent most of his adult life in Canada
Canada
. He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. He worked for 60 years at the University of Toronto
University of Toronto
and published twelve books . He was most noted for his work on regular polytopes and higher-dimensional geometries
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Tessellation
A TESSELLATION of a flat surface is the tiling of a plane using one or more geometric shapes , called tiles, with no overlaps and no gaps. In mathematics , tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and Semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups . A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons
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Coxeter
HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London
London
but spent most of his adult life in Canada
Canada
. He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. He worked for 60 years at the University of Toronto
University of Toronto
and published twelve books . He was most noted for his work on regular polytopes and higher-dimensional geometries
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George Olshevsky
GEORGE OLSHEVSKY (born 1946) is a freelance editor , writer , publisher , amateur paleontologist , and mathematician living in San Diego , California
California
. Olshevsky maintains the comprehensive online Dinosaur Genera List. He is known as the originator of the Birds Came First hypothesis in the descent of birds debate, which states that all dinosaurs are the descendants of small, arboreal and perhaps flying ancestors. Olshevsky is part of a collaborative effort to recognize and standardize terms used to describe uniform 4-polytopes and uniform polytopes , the analogues of uniform polyhedra in four and higher dimensions
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Special
SPECIAL or SPECIALS may refer to: CONTENTS * 1 Music * 2 Film and television * 3 Other uses * 4 See also MUSIC * Special (album) , a 1992 album by Vesta Williams * "Special" (Garbage song) , 1998 * "Special" (Mew song) , 2005 * "Special" (Stephen Lynch song) , 2000 * The Specials
The Specials
, a British band * "Special", a song by Violent Femmes on The Blind Leading the Naked * "Special", a song on
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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices A, B, and C is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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