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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(originally from Elte, 1912),[1] but it has now been superseded.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
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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.[1] 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)
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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's parallel postulate.[1] Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry
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On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences
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
Neil Sloane
while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009.[3] Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited
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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
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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
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Cartesian Coordinate System
A Cartesian coordinate system
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 (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0)
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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.[1] 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
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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)[1] is defined as A − B = c c + B ⊆ A . displaystyle A-B= mathbf c ,,mathbf c +Bsubseteq A
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC (February 9, 1907 – March 31, 2003)[2] 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. He was always called Donald, from his third name MacDonald.[3] 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.[2] 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
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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.[1] 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[2] (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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Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.[1][2] 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.Contents1 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 ReferencesEquivalent definitions of a closed set[edit] In 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. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. This is not to be confused with a closed manifold. Properties of closed sets[edit] A closed set contains its own boundary
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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
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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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