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Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k displaystyle u_ 0 ,dots ,u_ k in mathbb R ^ k are affinely independent, which means u 1 − u 0 , … , u k − u 0 displaystyle u_ 1 -u_ 0 ,dots ,u_ k -u_ 0 are linearly independent
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Simplicial Homology
In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology
Simplicial homology
arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead[1]). Simplicial homology
Simplicial homology
is defined by a simple recipe for any abstract simplicial complex
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Combinatorics
Combinatorics
Combinatorics
is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon.[1] According to H. J
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Power Of Two
In mathematics, a power of two is a number of the form 2n where n is an integer, i.e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to non-negative values,[1] so we have 1, 2, and 2 multiplied by itself a certain number of times.[2] Because two is the base of the binary numeral system, powers of two are common in computer science
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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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Tetrahedron Number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron
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Triangle Number
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n
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Hypercubic Honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols 4,3...3,4 and containing the symmetry of Coxeter
Coxeter
group Rn (or B~n-1) for n>=3. The tessellation is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope 3...3,4 . The hypercubic honeycombs are self-dual. Coxeter
Coxeter
named this family as δn+1 for an n-dimensional honeycomb. Wythoff construction
Wythoff construction
classes by dimension[edit] There are two general forms of the hypercube honeycombs, the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard. A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements
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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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Binomial Coefficient
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . displaystyle tbinom n k . It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula ( n k ) = n ! k ! ( n − k ) ! . displaystyle binom n k = frac n! k!(n-k)!
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Pieter Hendrik Schoute
Pieter Hendrik Schoute
Pieter Hendrik Schoute
(21 January 1846, Wormerveer
Wormerveer
– 18 April 1923, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. In 1886, he became member of the Royal Netherlands
Netherlands
Academy of Arts and Sciences.[1]Contents1 External links and references 2 References 3 Sources 4 External linksExternal links and references[edit]O'Connor, John J.; Robertson, Edmund F., "Pieter Hendrik Schoute", MacTutor History of Mathematics archive, University of St Andrews .References[edit]^ " Pieter Hendrik Schoute
Pieter Hendrik Schoute
(1846 - 1913)". Royal Netherlands
Netherlands
Academy of Arts and Sciences
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Henri Poincaré
Jules Henri Poincaré
Henri Poincaré
(French: [ɑ̃ʁi pwɛ̃kaʁe] ( listen);[2][3] 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist,"[4] since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.[5] He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory
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William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford
FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics,[1] geometry,[2] and computing.[3] Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry
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Finite Set
In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, 2 , 4 , 6 , 8 , 10 displaystyle 2,4,6,8,10 is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite: 1 , 2 , 3 , … . displaystyle 1,2,3,ldots . Finite sets are particularly important in combinatorics, the mathematical study of counting
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Abstract Simplicial Complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of non-empty finite sets closed under the operation of taking non-empty subsets.[1] In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.[2] An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra.Contents1 Definitions 2 Geometric realization 3 Examples 4 Enumeration 5 See also 6 ReferencesDefinitions[edit] A family Δ of non-empty finite subsets of a set S is an abstract simplicial complex if, for every set X in Δ, and every non-empty subset Y ⊂ X, Y also belongs to Δ. The finite sets that belong to Δ are called faces of the complex, and a face Y is said to belong to another face X if Y ⊂ X, so the definition of an abstract simplicial comp
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