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Polytopes
In elementary geometry, a polytope is a geometric object with Flat (geometry), flat sides (''Face (geometry), faces''). Polytopes are the generalization of three-dimensional polyhedron, 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 language, German term ''Polytop'' was coined by the mathematician Re ... [...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] |
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] |
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] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation 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 (an aperiodic set of prototiles). A '' tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sixteenth Stellation Of Icosahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra ( nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -dimensional simplex faces of the core -polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process. Construction The edge length of a great icosahedron is \frac times that of the original icosahedron. Images Formulas For a great icosahedron with edge length E (the edge of its dodecahedron core), \text = \frac \text = \frac \text = \frac \text = 3\sqrt(5+4\sqrt)\text^2 \text = \text^3 As ... [...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] |
Reinhold Hoppe
Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the University of Berlin. Education and career Hoppe was a student of Johann August Grunert at the University of Greifswald, graduating in 1842 and becoming an English and mathematics teacher. He completed his doctorate in 1850 in Halle and his habilitation in mathematics in 1853 in Berlin under Peter Gustav Lejeune Dirichlet. He also tried to obtain a habilitation in philosophy at the same time, but was denied until a later re-application in 1871. He worked at Berlin as a privatdozent, and then after 1870 as a professor, but with few students and little remuneration. When Grunert died in 1872, Hoppe took over the editorship of the mathematical journal founded by Grunert, the ''Archiv der Mathematik und Physik''. Hoppe in turn continued as editor until his own death, in 1900.. See in particulapp. 435–437 In 1890, Hoppe was one of the 31 founding members of the G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Polyhedra
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called '' spherical polygons''. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron. The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron. History During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Stellation Of Icosahedron
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric to each other. Branko Grünbaum has conjectured that it is the only Euclidean isohedron with convex faces of six or more sides, but the small hexagonal hexecontahedron is another example. Geometry The faces are equilateral hexagons, with alternating angles of \arccos(-\frac)\approx 104.477\,512\,185\,93^ and \arccos(\frac)+60^\approx 135.522\,487\,814\,07^. The dihedral angle equals \arccos(-\frac)\approx 109.471\,220\,634\,49. Related shapes The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron. (1st Edn Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
