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Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a ''realization'' of an abstract polytope in some real N-dimensional space, typically Euclidean space, Euclidean. This abstract definition allows more general combinatorics, combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory. Introductory concepts Traditional versus abstract polytopes In Euclidean geometry, two shapes that are not Similar (geometry), similar can nonetheless share a common structure. For example, a square and a trapezoid both comprise an alternating chain of four vertex (geometry), vertices and four sides, which makes them quadrilaterals. They are said to be isomorphic or “structure preserving”. This common structure may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (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 identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face (geometry)
In solid geometry, a face is a flat surface (a Plane (geometry), planar region (mathematics), region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane ''tessellation, tile''. For example, any of the six square (geometry), 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. With this meaning, the 4-dimensional tesseract has 24 square faces, each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and for his generalization of Descartes' theorem on tangent circles to four and higher dimensions. Biography Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs,UK Census 1871, RG10-863-89-23 and his wife Eleanor Gosset (formerly Thorold). He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. In 1900 he married Emily Florence Wood, and they subsequently had two children, named Kathleen and John.UK Census 1911, RG14-181-9123-19 Mathematics According to H. S. M. Coxeter, after obtaining his law degree in 1896 and having no client ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon, nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular polyhedron, regular, Uniform polyhedron, uniform polyhedra, or the Dual polyhedron, duals of the uniform polyhedra. All these stars are of the self-intersecting kind. Self-intersecting star polyhedra Regular star polyhedra The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting Face (geometry), faces, or self-intersecting vertex figures. There are four List of regular polytopes and compounds#Three dimensions 2, regular star polyhedra, known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polytope
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind. Self-intersecting star polyhedra Regular star polyhedra The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures. There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The Schläfli symbol implies faces with ''p'' sides, and vertex figures with ''q'' sides. Two of them have pent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many Facet (geometry), facets. Definition Abstract apeirotope An Abstract polytope, abstract ''n''-polytope is a partially ordered set ''P'' (whose elements are called ''faces'') such that ''P'' contains a least face and a greatest face, each maximal totally ordered subset (called a ''flag'') contains exactly ''n'' + 2 faces, ''P'' is strongly connected, and there are exactly two faces that lie strictly between ''a'' and ''b'' are two faces whose ranks differ by two. An abstract polytope is called an abstract apeirotope if it has infinitely many faces. An abstract polytope is called ''regular'' if its automorphism group Γ(''P'') acts transitively on all of the flags of ''P''. Classification There are two main geometric classes of apeirotope: *honeycomb (geometry), honeycombs in ''n'' dimensions, which completely fill an n-dimensional space, ''n''-dimensional space. *skew apeirotopes, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alicia Boole Stott
Alicia Boole Stott (8 June 1860 – 17 December 1940) was a British mathematician. She made a number of contributions to the field and was awarded an honorary doctorate from the University of Groningen. She grasped four-dimensional geometry from an early age, and introduced the term "polytope" for a convex solid in four or more dimensions. Personal life Alicia Boole was born in Cork, Ireland, the third of five daughters of English parents: the mathematician and logician George Boole and Mary Everest Boole, a self-taught mathematician and educationalist. Of her sisters, Lucy Everest Boole was a chemist and pharmacist and Ethel Lilian Voynich was a novelist. After her father's sudden death in 1864, the family moved to London, where her mother became the librarian at Queen's College, London. Alicia attended the school attached to Queens' College with one of her sisters, but never attended university. She was known to her friends and family as Alice, though she always published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely Combinatorics, combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex., Section 4.3 Definitions A simplicial complex \mathcal is a set of Simplex, simplices that satisfies the following conditions: # Every Simplex#Elements, face of a simplex from \mathcal is also in \mathcal. # The non-empty Set intersection, intersection of any two simplices \sigma_1, \sigma_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
