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Hypercube
In geometry , a HYPERCUBE is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed , compact , convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions , perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimension is equal to n {displaystyle {sqrt {n}}} . An n-dimensional hypercube is also called an N-CUBE or an N-DIMENSIONAL CUBE. The term "measure polytope" is also used, notably in the work of H. S. M. Coxeter
Coxeter
(originally from Elte, 1912), but it has now been superseded. The hypercube is the special case of a hyperrectangle (also called an n-orthotope). A UNIT HYPERCUBE is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called "THE" UNIT HYPERCUBE
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Minkowski Sum
In geometry , the MINKOWSKI SUM (also known as dilation ) of two sets of position vectors A and B in Euclidean space
Euclidean space
is formed by adding each vector in A to each vector in B, i.e., the set A + B = { a + b a A , b B } . {displaystyle A+B={mathbf {a} +mathbf {b} ,,mathbf {a} in A, mathbf {b} in B}.} Analogously, the MINKOWSKI DIFFERENCE (or geometric difference) is defined as A B = { c c + B A } . {displaystyle A-B={mathbf {c} ,,mathbf {c} +Bsubseteq A}.} It is important to note that in general A B A + ( B ) {displaystyle A-Bneq A+(-B)} . For instance, in a one-dimensional case A = {displaystyle A=} and B = {displaystyle B=} the Minkowski difference A B = {displaystyle A-B=} , whereas A + ( B ) = A + B =
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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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Perpendicular
In elementary geometry , the property of being PERPENDICULAR (PERPENDICULARITY) is the relationship between two lines which meet at a right angle (90 degrees ). The property extends to other related geometric objects . A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles . Perpendicularity can be shown to be symmetric , meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. Perpendicularity easily extends to segments and rays
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Line Segment
In geometry , a LINE SEGMENT is a part of a line that is bounded by two distinct end points , and contains every point on the line between its endpoints. A CLOSED LINE SEGMENT includes both endpoints, while an OPEN LINE SEGMENT excludes both endpoints; a HALF-OPEN LINE SEGMENT includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or otherwise a diagonal . When the end points both lie on a curve such as a circle , a line segment is called a chord (of that curve)
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Convex Hull
In mathematics , the CONVEX HULL or CONVEX ENVELOPE of a set X of points in the Euclidean plane
Euclidean plane
or in a Euclidean space
Euclidean space
(or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids . The algorithmic problem of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry
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Cartesian Coordinate System
A CARTESIAN COORDINATE SYSTEM is a coordinate system that specifies each point uniquely in a plane by a pair of numerical COORDINATES, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin
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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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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
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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On-Line Encyclopedia Of Integer Sequences
The ON-LINE ENCYCLOPEDIA OF INTEGER SEQUENCES (OEIS), also cited simply as SLOANE\'S, is an online database of integer sequences . It was created and maintained by Neil Sloane while a researcher at AT&T Labs . Foreseeing his retirement from AT"> it contains nearly 280,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords , mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword and by subsequence
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Recurrence Relation
In mathematics , a RECURRENCE RELATION is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms. The term DIFFERENCE EQUATION sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation
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Parallel (geometry)
In geometry , PARALLEL lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space
Euclidean space
that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines . Parallel planes are planes in the same three-dimensional space that never meet. Parallel lines are the subject of Euclid
Euclid
's parallel postulate . Parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as hyperbolic space , have analogous properties that are sometimes referred to as parallelism
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Factorial
In mathematics , the FACTORIAL of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5 ! = 5 4 3 2 1 = 120. {displaystyle 5!=5times 4times 3times 2times 1=120.} The value of 0! is 1, according to the convention for an empty product . The factorial operation is encountered in many areas of mathematics, notably in combinatorics , algebra , and mathematical analysis . Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). This fact was known at least as early as the 12th century, to Indian scholars. Fabian Stedman , in 1677, described factorials as applied to change ringing
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Perspective Projection
PERSPECTIVE (from Latin : perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight. Italian Renaissance
Italian Renaissance
painters and architects including Filippo Brunelleschi , Masaccio
Masaccio
, Paolo Uccello
Paolo Uccello
, Piero della Francesca
Piero della Francesca
and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks, thus contributing to the mathematics of art
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Closed Set
In geometry , topology , and related branches of mathematics , a CLOSED SET is a set whose complement is an open set . In a topological space , a closed set can be defined as a set which contains all its limit points . In a complete metric space , a closed set is a set which is closed under the limit operation. CONTENTS * 1 Equivalent definitions of a closed set * 2 Properties of closed sets * 3 Examples of closed sets * 4 More about closed sets * 5 See also * 6 References EQUIVALENT DEFINITIONS OF A CLOSED SETIn a topological space , a set is CLOSED if and only if it coincides with its closure . Equivalently, a set is closed if and only if it contains all of its limit points . This is not to be confused with a closed manifold . PROPERTIES OF CLOSED SETSA closed set contains its own boundary . In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set
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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
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry
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
Euclid
, whose treatment, Euclid\'s Elements , set a standard for many centuries to follow. Geometry
Geometry
arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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