1.
Polytrope
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The constant n is known as the polytropic index. The polytropic exponent γ = /n is sometimes preferred over n and this relation need not be interpreted as an equation of state, although a gas following such an equation of state does produce a polytropic solution to the Lane–Emden equation. Rather, this is simply a relation that expresses an assumption about the change of pressure with radius in terms of the change of density with radius, yielding a solution to the Lane–Emden equation. Sometimes the word polytrope may be used to refer to an equation of state that similar to the thermodynamic relation above. It is preferable to refer to the fluid itself as a polytropic fluid, the equation of state of a polytropic fluid is general enough that such idealized fluids find wide use outside of the limited problem of polytropes. The polytropic index has shown to be equivalent to the pressure derivative of the bulk modulus where its relation to the Murnaghan equation of state has also been demonstrated. Neutron stars are well modeled by polytropes with index between n =0.5 and n =1, a polytrope with index n =1.5 is a good model for fully convective star cores, brown dwarfs, giant gaseous planets, or even for rocky planets. A polytrope with index n =5 has an infinite radius and it corresponds to the simplest plausible model of a self-consistent stellar system, first studied by A. Schuster in 1883. In general as the index increases, the density distribution is more heavily weighted toward the center of the body. Polytropic process Equation of state Murnaghan equation of state
2.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
3.
Geometry
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Geometry 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 arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
4.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
5.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. 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 set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
6.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
7.
Spherical polyhedron
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In mathematics, 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. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
8.
Abstract polytope
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An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, the term polytope is a generalisation of polygons and polyhedra into any number of dimensions. In Euclidean geometry, the six quadrilaterals illustrated are all different, yet they have a common structure in the alternating chain of four vertices and four sides which gives them their name. They are said to be isomorphic or “structure preserving”, the measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope. What is true for traditional polytopes may not be so for abstract ones, for example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope. A traditional geometric polytope is said to be a realisation of the abstract polytope. A realisation is a mapping or injection of the object into a real space, typically Euclidean. The six quadrilaterals shown are all distinct realisations of the abstract quadrilateral, some of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realisations. A conventional polytope is a faithful realisation, in an abstract polytope, each structural element - vertex, edge, cell, etc. is associated with a corresponding member or element of the set. The term face often refers to any such element e. g. a vertex, edge or a general k-face, the faces are ranked according to their associated real dimension, vertices have rank =0, edges rank =1 and so on. This usage of incidence also occurs in Finite geometry, although it differs from traditional geometry, for example in the square abcd, edges ab and bc are not abstractly incident. A polytope is defined as a set of faces P with an order relation <. Formally, P will be an ordered set, or poset. Just as the zero is necessary in mathematics, so also set theory requires an empty set which, technically. In an abstract polytope this is known as the least or null face and is a subface of all the others, since the least face is one level below the vertices or 0-faces, its rank is −1 and may be denoted as F−1. There is also a face of which all the others are subfaces. This is called the greatest face, in an n-dimensional polytope, the greatest face has rank = n and may be denoted as Fn. It is sometimes realized as the interior of the geometric figure and these least and greatest faces are sometimes called improper faces, with all others being proper faces
9.
German language
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German is a West Germanic language that is mainly spoken in Central Europe. It is the most widely spoken and official language in Germany, Austria, Switzerland, South Tyrol, the German-speaking Community of Belgium and it is also one of the three official languages of Luxembourg. Major languages which are most similar to German include other members of the West Germanic language branch, such as Afrikaans, Dutch, English, Luxembourgish and it is the second most widely spoken Germanic language, after English. One of the languages of the world, German is the first language of about 95 million people worldwide. The German speaking countries are ranked fifth in terms of publication of new books. German derives most of its vocabulary from the Germanic branch of the Indo-European language family, a portion of German words are derived from Latin and Greek, and fewer are borrowed from French and English. With slightly different standardized variants, German is a pluricentric language, like English, German is also notable for its broad spectrum of dialects, with many unique varieties existing in Europe and also other parts of the world. The history of the German language begins with the High German consonant shift during the migration period, when Martin Luther translated the Bible, he based his translation primarily on the standard bureaucratic language used in Saxony, also known as Meißner Deutsch. Copies of Luthers Bible featured a long list of glosses for each region that translated words which were unknown in the region into the regional dialect. Roman Catholics initially rejected Luthers translation, and tried to create their own Catholic standard of the German language – the difference in relation to Protestant German was minimal. It was not until the middle of the 18th century that a widely accepted standard was created, until about 1800, standard German was mainly a written language, in urban northern Germany, the local Low German dialects were spoken. Standard German, which was different, was often learned as a foreign language with uncertain pronunciation. Northern German pronunciation was considered the standard in prescriptive pronunciation guides though, however, German was the language of commerce and government in the Habsburg Empire, which encompassed a large area of Central and Eastern Europe. Until the mid-19th century, it was essentially the language of townspeople throughout most of the Empire and its use indicated that the speaker was a merchant or someone from an urban area, regardless of nationality. Some cities, such as Prague and Budapest, were gradually Germanized in the years after their incorporation into the Habsburg domain, others, such as Pozsony, were originally settled during the Habsburg period, and were primarily German at that time. Prague, Budapest and Bratislava as well as cities like Zagreb, the most comprehensive guide to the vocabulary of the German language is found within the Deutsches Wörterbuch. This dictionary was created by the Brothers Grimm and is composed of 16 parts which were issued between 1852 and 1860, in 1872, grammatical and orthographic rules first appeared in the Duden Handbook. In 1901, the 2nd Orthographical Conference ended with a standardization of the German language in its written form
10.
Alicia Boole Stott
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Alicia Boole Stott was an Irish-English mathematician. Despite never holding a position, she made a number of valuable contributions to the field. She is best known for coining the term polytope for a solid in four dimensions. Alicia Boole was born in Cork, Ireland, the daughter of mathematician and logician George Boole and Mary Everest Boole. Many of her sisters were notable in their own right, lucy Everest Boole was a chemist and pharmacist and Ethel Lilian Voynich was a novelist. After her fathers death in 1864, the family moved to London. She attended the school attached to Queens College with her sister and she was first exposed to geometric models by her brother-in-law Charles Howard Hinton when she was just 18, and developed the ability to visualise in a fourth dimension. She found that there were exactly six regular polytopes in four dimensions and she made cardboard models of all these sections. She coined the term polytopes to describe them, after taking up secretarial work near Liverpool in 1889 she met and married Walter Stott, an actuary, in 1890. They had two children together, Mary and Leonard, Stott learned of Pieter Schoutes work on central sections of the regular polytopes in 1895. Schoute came to England and worked with Alicia Stott, persuading her to publish her results which she did in two published in Amsterdam in 1900 and 1910. The University of Groningen honoured her by inviting her to attend the celebrations of the university. After Schoutes death in 1923 Alicia took a hiatus from mathematical work, in 1930 she was introduced by her nephew Geoffrey Ingram Taylor to H. S. M. Coxeter and they worked together on various problems, Alicia made two further important discoveries relating to constructions for polyhedra related to the golden section. She presented a joint paper with Coxeter at Cambridge University, Coxeter later wrote, The strength and simplicity of her character combined with the diversity of her interests to make her an inspiring friend. Alicia died in Middlesex in 1940, in Spring 2001 a paper roll of coloured drawings of polyhedra was found at Groningen University. Though unsigned, it was recognised as Alicias work. It led to research by Irene Polo-Blanco, Polo-Blanco dedicated a chapter to Alicias work in her book Theory and History of Mathematical Models
11.
Convex polytope
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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. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -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, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
12.
Euler characteristic
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It is commonly denoted by χ. The Euler characteristic was originally defined for polyhedra and used to prove theorems about them. Leonhard Euler, for whom the concept is named, was responsible for much of early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, any convex polyhedrons surface has Euler characteristic V − E + F =2. This equation is known as Eulers polyhedron formula and it corresponds to the Euler characteristic of the sphere, and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below and this version holds both for convex polyhedra and the non-convex Kepler-Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0. The Euler characteristic can be defined for connected plane graphs by the same V − E + F formula as for polyhedral surfaces, the Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of determined by G. For trees, E = V −1 and F =1, if G has C components, the same argument by induction on F shows that V − E + F − C =1. One of the few graph theory papers of Cauchy also proves this result, via stereographic projection the plane maps to the two-dimensional sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchys proof of Eulers formula given below, there are many proofs of Eulers formula. One was given by Cauchy in 1811, as follows and it applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface, after this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, therefore, proving Eulers formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that arent connected yet. This adds one edge and one face and does not change the number of vertices, continue adding edges in this manner until all of the faces are triangular. This decreases the number of edges and faces by one each and does not change the number of vertices, remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph
13.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
14.
Simplicial complex
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In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. A simplicial complex K is a set of simplices that satisfies the conditions,1. Any face of a simplex from K is also in K.2, the intersection of any two simplices σ1, σ2 ∈ K is either ∅ or a face of both σ1 and σ2. Note that the empty set is a face of every simplex, see also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex K is a complex where the largest dimension of any simplex in K equals k. For instance, a simplicial 2-complex must contain at least one triangle, a pure or homogeneous simplicial k-complex K is a simplicial complex where every simplex of dimension less than k is a face of some simplex σ ∈ K of dimension exactly k. Informally, a pure 1-complex looks like its made of a bunch of lines, an example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. A facet is any simplex in a complex that is not a face of any larger simplex, a pure simplicial complex can be thought of as a complex where all facets have the same dimension. Sometimes the term face is used to refer to a simplex of a complex, for a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is used in a broader sense to denote a set homeomorphic to a simplex. The underlying space, sometimes called the carrier of a complex is the union of its simplices. Let K be a complex and let S be a collection of simplices in K. The closure of S is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S is obtained by adding to S each face of every simplex in S. The star of S is the union of the stars of each simplex in S, for a single simplex s, the star of s is the set of simplices having a face in s. The link of S equals Cl St S − St Cl S and it is the closed star of S minus the stars of all faces of S. In algebraic topology, simplicial complexes are useful for concrete calculations
15.
Simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
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Star polyhedron
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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 kinds of star polyhedron, Polyhedra which self-intersect in a repetitive way. Concave polyhedra of a 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, 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 pentagrammic faces and two have pentagrammic vertex figures. These images show each form with a single face colored yellow to show the visible portion of that face, There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms, and their duals. The uniform and dual uniform polyhedra are also self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures or both, the uniform star polyhedra have regular faces or regular star polygon faces. The dual uniform polyhedra have regular faces or regular star polygon vertex figures. Beyond the forms above, there are unlimited classes of self-intersecting polyhedra, two important classes are the stellations of convex polyhedra and their duals, the facettings of the dual polyhedra. For example, the complete stellation of the icosahedron can be interpreted as a polyhedron composed of 12 identical faces. Below is an illustration of this polyhedron with one drawn in yellow. A similarly self-intersecting polytopes in any number of dimensions is called a star polytope, a regular polytope is a star polytope if either its facet or its vertex figure is a star polytope. In four dimensions, the 10 regular star polychora are called the Schläfli-Hess polychora, analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler-Poinsot polyhedra. For example, the grand stellated 120-cell, projected orthogonally into 3-space, looks like this. A polyhedron which does not cross itself, such that all of the interior can be seen from one point, is an example of a star domain
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Spherical tiling
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In mathematics, 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. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
18.
Elliptic space
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a line must intersect. In fact, the perpendiculars on one side all intersect at the pole of the given line. There are no points in elliptic geometry. Every point corresponds to a polar line of which it is the absolute pole. Any point on this line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant, the distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name elliptic is possibly misleading and it does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes, analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity. A simple way to picture elliptic geometry is to look at a globe, neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles. More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, with this identification of antipodal points, the model satisfies Euclids first postulate, which states that two points uniquely determine a line. Metaphorically, we can imagine geometers who are like living on the surface of a sphere. Even if the ants are unable to move off the surface, they can still construct lines, the existence of a third dimension is irrelevant to the ants ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the axioms is incapable of expressing the distinction between one model and another. In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the figures are similar, i. e. they have the same angles. In elliptic geometry this is not the case, for example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere. A line segment therefore cannot be scaled up indefinitely, a geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space
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Toroid
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In mathematics, a toroid is a surface of revolution with a hole in the middle, like a doughnut. The axis of revolution passes through the hole and so does not intersect the surface, for example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus, the term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes, a g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2, annulus The dictionary definition of toroid at Wiktionary
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Projective polyhedron
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In geometry, a projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere –, however, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra. As cellular decompositions of the plane, they have Euler characteristic 1. The qualifier globally is to contrast with locally projective polyhedra, which are defined in the theory of abstract polyhedra, non-overlapping projective polyhedra correspond to spherical polyhedra with central symmetry. This is elaborated and extended below in relation with spherical polyhedra, on the other hand, the tetrahedron does not have central symmetry, so there is no hemi-tetrahedron. See relation with spherical polyhedra below on how the tetrahedron is treated, note that the prefix hemi- is also used to refer to hemipolyhedra, which are uniform polyhedra having some faces that pass through the center of symmetry. As these do not define spherical polyhedra, they do not define projective polyhedra by the quotient map from 3-space to the projective plane and it is 2-covered by the cuboctahedron, and can be realized as the quotient of the spherical cuboctahedron by the antipodal map. It is the uniform polyhedron that is projective – that is. Further, because a covering map is a homeomorphism, both the spherical and the corresponding projective polyhedra have the same abstract vertex figure. For example, the 2-fold cover of the hemi-cube is the cube.4.4 vertex figure, see symmetry group below for elaboration and other dimensions. Spherical polyhedra without central symmetry do not define a polyhedron, as the images of vertices, edges. This geometrizes the Galois connection at the level of finite subgroups of O and PO, for example, the tetrahedron is not centrally symmetric, and has 4 vertices,6 edges, and 4 faces, and vertex figure 3.3.3. Its image in the plane has 4 vertices,6 edges. The cover of this is the stellated octahedron – equivalently, the compound of two tetrahedra – which has 8 vertices,12 edges, and 8 faces, and vertex figure 3.3.3. In the context of abstract polytopes, one refers to locally projective polytopes – see Abstract polytope. Projective polytopes can be defined in higher dimension as tessellations of space in one less dimension. If n=2k is even, then O = SO× decomposes as a product, the projective plane is non-orientable, and thus there is no distinct notion of orientation-preserving isometries of a projective polyhedron, which is reflected in the equality PSO = PO. Further, in odd projective dimension P S O ≠ P O and is instead a proper subgroup, so there is a distinct notion of orientation-preserving isometries
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Toroidal polyhedron
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In geometry, a toroidal polyhedron is a polyhedron which is also a toroid, having a topological genus, g, of 1 or greater. The Euler characteristic, χ=V+F-E, of a toroid is 2. Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by two polygons, and the link of each vertex should be a single cycle that alternates between the edges and polygons that meet at that vertex. For toroidal polyhedra, this manifold be an orientable surface, some authors restrict the phrase toroidal polyhedra to mean more specifically polyhedra topologically equivalent to the torus. Intermediate between these two extremes are immersed toroidal polyhedra, polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to each other. In all of these cases the nature of a polyhedron can be verified by its orientability. Two of the simplest possible embedded toroidal polyhedra are the Császár, the Császár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces. It and the tetrahedron are the only known polyhedra in which every line segment connecting two vertices forms an edge of the polyhedron. The Császár polyhedron has the fewest possible vertices of any embedded toroidal polyhedron, and these are called Stewart toroids, named after Professor Bonnie Stewart who explored their existence. They are analogous to the Johnson solids in the case of convex polyhedra and they include also toroidal deltahedra, polyhedra whose faces are all equilateral triangles. A restricted class of Stewart toroids, also defined by Stewart, are the quasi-convex toroidal polyhedra and these are Stewart toroids that include all of the edges of their convex hulls. For such a polyhedron, each face of the convex hull either lies on the surface of the toroid, examples include the genus-1 octahemioctahedron, the genus-3 small cubicuboctahedron, and the genus-4 great dodecahedron. A crown polyhedron or stephanoid is a polyhedron which is also noble. Crown polyhedra are self-intersecting and topologically self-dual, infinite skew polyhedron Projective polyhedron Spherical polyhedron Toroidal graph Weisstein, Eric W. Toroidal polyhedron. Stewart Toroids Stewarts polyhedra Toroidal Polyhedra Stewart toroids
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Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
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4-polytope
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In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements, vertices, edges, faces, each face is shared by exactly two cells. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron, topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space, similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space, a 4-polytope is a closed four-dimensional figure. It comprises vertices, edges, faces and cells, a cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i. e. it is not a compound, the most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. 4-polytopes cannot be seen in space due to their extra dimension. Several techniques are used to help visualise them, Orthogonal projection Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes. Perspective projection Just as a 3D shape can be projected onto a flat sheet, sectioning Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut hypersurface in three dimensions. A sequence of sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce an animation of these cross sections. The topology of any given 4-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, like all polytopes, 4-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the shapes of the non-convex star polygons. A 4-polytope is regular if it is transitive on its flags and this means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron
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Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
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Polytope
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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli, the German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a class of objects. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes and they represent different approaches to generalizing the convex polytopes to include other objects with similar properties. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold, an example of this approach defines a polytope as a set of points that admits a simplicial decomposition. However this definition does not allow star polytopes with interior structures, the discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra and this approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms polytope and polyhedron are used in a different sense and this terminology is typically confined to polytopes and polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells, terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically, authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element, the terms adopted in this article are given in the table below, An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope, Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope and these bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point, a 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, the convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite, Polytopes are defined in this way, e. g. in linear programming
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Vertex (geometry)
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In geometry, a vertex 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. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
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Bounded set
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Bounded and boundary are distinct concepts, for the latter see boundary. A circle in isolation is a bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, conversely, a set which is not bounded is called unbounded. The word bounded makes no sense in a topological space without a metric. A set S of real numbers is called bounded from above if there is a number k such that k ≥ s for all s in S. The number k is called a bound of S. The terms bounded from below and lower bound are similarly defined, a set S is bounded if it has both upper and lower bounds. Therefore, a set of numbers is bounded if it is contained in a finite interval. M is a metric space if M is bounded as a subset of itself. For subsets of Rn the two are equivalent, a metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space Rn is compact if and only if it is closed and bounded, in topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces. A set of numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any ordered set. Note that this general concept of boundedness does not correspond to a notion of size. A subset S of an ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called a bound of S. The concepts of bounded below and lower bound are defined similarly, a subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval
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Convex body
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In mathematics, a convex body in n-dimensional Euclidean space R n is a compact convex set with non-empty interior. Important examples of convex bodies are the Euclidean ball, the hypercube, list of convexity topics Gardner, Richard J
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Convex hull
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In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space is the smallest convex set that contains 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 set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry. A set of points is defined to be if it contains the line segments connecting each pair of its points. The convex hull of a given set X may be defined as The minimal convex set containing X The intersection of all convex sets containing X The set of all combinations of points in X. The union of all simplices with vertices in X and it is not obvious that the first definition makes sense, why should there exist a unique minimal convex set containing X, for every X. Thus, it is exactly the unique minimal convex set containing X. Each convex set containing X must contain all convex combinations of points in X, in fact, according to Carathéodorys theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N +1 points are sufficient in the definition above. If the convex hull of X is a set, then it is the intersection of all closed half-spaces containing X. The hyperplane separation theorem proves that in case, each point not in the convex hull can be separated from the convex hull by a half-space. However, there exist convex sets, and convex hulls of sets, more abstractly, the convex-hull operator Conv has the characteristic properties of a closure operator, It is extensive, meaning that the convex hull of every set X is a superset of X. It is non-decreasing, meaning that, for two sets X and Y with X ⊆ Y, the convex hull of X is a subset of the convex hull of Y. It is idempotent, meaning that for every X, the hull of the convex hull of X is the same as the convex hull of X. The convex hull of a point set S is the set of all convex combinations of its points. For each choice of coefficients, the convex combination is a point in the convex hull. Expressing this as a formula, the convex hull is the set. The convex hull of a point set S ⊊ R n forms a convex polygon when n =2. Each point x i in S that is not in the hull of the other points is called a vertex of Conv . In fact, every convex polytope in R n is the hull of its vertices
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H. S. M. Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in 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 he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
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Cell (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
32.
Linear programming
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Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Linear programming is a case of mathematical programming. More formally, linear programming is a technique for the optimization of an objective function, subject to linear equality. Its feasible region is a polytope, which is a set defined as the intersection of finitely many half spaces. Its objective function is an affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest value if such a point exists, the expression to be maximized or minimized is called the objective function. The inequalities Ax ≤ b and x ≥0 are the constraints which specify a convex polytope over which the function is to be optimized. In this context, two vectors are comparable when they have the same dimensions, if every entry in the first is less-than or equal-to the corresponding entry in the second then we can say the first vector is less-than or equal-to the second vector. Linear programming can be applied to fields of study. It is widely used in business and economics, and is utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications and it has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design. The first linear programming formulation of a problem that is equivalent to the linear programming problem was given by Leonid Kantorovich in 1939. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army, about the same time as Kantorovich, the Dutch-American economist T. C. Koopmans formulated classical economic problems as linear programs, Kantorovich and Koopmans later shared the 1975 Nobel prize in economics. Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force, in 1947, Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases. Dantzig provided formal proof in an unpublished report A Theorem on Linear Inequalities on January 5,1948, postwar, many industries found its use in their daily planning. Dantzigs original example was to find the best assignment of 70 people to 70 jobs, the computing power required to test all the permutations to select the best assignment is vast, the number of possible configurations exceeds the number of particles in the observable universe. However, it only a moment to find the optimum solution by posing the problem as a linear program
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Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
34.
Regular polytope
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to
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Flag (geometry)
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In geometry, a flag is a sequence of faces of a polytope, each contained in the next, with just one face from each dimension. More formally, a flag ψ of an n-polytope is a set such that Fi ≤ Fi+1, since, however, the minimal face F−1 and the maximal face Fn must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces, for example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces. A flag of a polyhedron is called a dart. A polytope may be regarded as if, and only if. This level of abstraction generalizes both the concept given above as well as the related flag concept from linear algebra. A flag is maximal if it is not contained in a larger flag, when all maximal flags of an incidence geometry have the same size, this common value is the rank of the geometry
36.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
37.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
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Hypercube
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In geometry, a hypercube is an n-dimensional analogue of a square and a cube. A unit hypercubes longest diagonal in n-dimensions is equal to 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, the hypercube is the special case of a hyperrectangle. A unit hypercube is a hypercube whose side has one unit. Often, the hypercube whose corners are the 2n points in Rn with coordinates equal to 0 or 1 is called the unit hypercube, a hypercube can be defined by increasing the numbers of dimensions of a shape,0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment,2 – If one moves this line segment its length in a perpendicular direction from itself, it sweeps out a 2-dimensional square. 3 – If one moves the square one unit length in the perpendicular to the plane it lies on. 4 – If one moves the cube one unit length into the fourth dimension and this can be generalized to any number of dimensions. The 1-skeleton of a hypercube is a hypercube graph, a unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates. It has a length of 1 and an n-dimensional volume of 1. An n-dimensional hypercube is also regarded as the convex hull of all sign permutations of the coordinates. This form is chosen due to ease of writing out the coordinates. Its edge length is 2, and its volume is 2n. Every n-cube of n >0 is composed of elements, or n-cubes of a dimension, on the -dimensional surface on the parent hypercube. A side is any element of -dimension of the parent hypercube, a hypercube of dimension n has 2n sides. The number of vertices of a hypercube is 2 n, the number of m-dimensional hypercubes on the boundary of an n-cube is E m, n =2 n − m, where = n. m. and n. denotes the factorial of n. For example, the boundary of a 4-cube contains 8 cubes,24 squares,32 lines and 16 vertices and this identity can be proved by combinatorial arguments, each of the 2 n vertices defines a vertex in a m-dimensional boundary. There are ways of choosing which lines that defines the subspace that the boundary is in, but, each side is counted 2 m times since it has that many vertices, we need to divide with this number
39.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
40.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
41.
Cross-polytope
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In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is an octahedron. Its facets are simplexes of the dimension, while the cross-polytopes vertex figure is another cross-polytope from the previous dimension. The vertices of a cross-polytope are all the permutations of, the cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the unit ball in the ℓ1-norm on Rn. In 1 dimension the cross-polytope is simply the line segment, in 2 dimensions it is a square with vertices, in 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these, the cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T, the 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six convex regular 4-polytopes and these 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. The cross polytope family is one of three regular polytope families, labeled by Coxeter as βn, the two being the hypercube family, labeled as γn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn, the n-dimensional cross-polytope has 2n vertices, and 2n facets all of which are n−1 simplices. The vertex figures are all n −1 cross-polytopes, the Schläfli symbol of the cross-polytope is. The dihedral angle of the n-dimensional cross-polytope is δ n = arccos and this gives, δ2 = arccos = 90°, δ3 = arccos =109. 47°, δ4 = arccos = 120°, δ5 = arccos =126. 87°. The volume of the n-dimensional cross-polytope is 2 n n. Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2-gon petrie polygon of the dimension, seen as a bipyramid, projected down the axis. The vertices of a cross polytope are all at equal distance from each other in the Manhattan distance. Kusners conjecture states that this set of 2d points is the largest possible equidistant set for this distance, Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes, βpn =22. 2p, or. Real solutions exist with p=2, i. e. β2n = βn =22.22 =, for p>2, they exist in C n
42.
Regular octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size