1.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
2.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves
3.
The Fifty-Nine Icosahedra
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The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie and it enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition by Springer-Verlag followed in 1982, K. and D. Crennell completely reset the text and redrew the plates and diagrams for Tarquins 1999 Third Edition, also adding new reference material and photographs. Although Miller did not contribute to the book directly, he was a colleague of Coxeter. All parts composing the faces must be the same in each plane, the parts included in any one plane must have trigonal symmetry, without or with reflection. This secures icosahedral symmetry for the whole solid, the parts included in any plane must all be accessible in the completed solid. We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure, but we allow the combination of an enantiomorphous pair having no common part. Rules to are symmetry requirements for the face planes, rule excludes buried holes, to ensure that no two stellations look outwardly identical. Rule prevents any disconnected compound of simpler stellations, Coxeter was the main driving force behind the work. He carried out the analysis based on Millers rules, adopting a number of techniques such as combinatorics. He observed that the stellation diagram comprised many line segments and he then developed procedures for manipulating combinations of the adjacent plane regions, to formally enumerate the combinations allowed under Millers rules. His graph, reproduced here, shows the connectivity of the various faces identified in the stellation diagram, based on this he tested all possible combinations against Millers rules, confirming the result of Coxeters more analytical approach. Flathers contribution was indirect, he made models of all 59. When he first met Coxeter he had made many stellations. He went on to complete the series of fifty-nine, which are preserved in the library of Cambridge University. The library also holds some non-Miller models, but it is not known whether these were made by Flather or by Millers later students, john Flinders Petrie was a lifelong friend of Coxeter and had a remarkable ability to visualise four-dimensional geometry. He and Coxeter had worked together on many mathematical problems and his direct contribution to the fifty nine icosahedra was the exquisite set of three-dimensional drawings which provide much of the fascination of the published work. For the Third Edition, Kate and David Crennell completely reset the text and redrew the illustrations and they also added a reference section containing tables, diagrams, and photographs of some of the Cambridge models
4.
Wenninger
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Father Magnus J. Wenninger OSB was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Born to German immigrants in Park Falls, Wisconsin, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, the ad was for a preparatory school in Collegeville, Minnesota, associated with the Benedictine St. John’s University. While admitting to feeling homesick at first, Wenninger quickly made friends and, after a year, knew that this was where he needed to be. He was a student in a section of the school that functioned as a “minor seminary” – later moving on into St. John’s where he studied philosophy and theology. Wenninger became a Benedictine monk, he took on his monastic name Magnus, at the start of his career, Wenninger did not set out on a path one might expect would lead to his becoming the great polyhedronist that he is known as today. Rather, a few chance happenings and seemingly minor decisions shaped a course for Wenninger that led to his groundbreaking studies, shortly after becoming a priest, Wenninger’s Abbot informed him that their order was starting up a school in the Bahamas. It was decided that Wenninger would be assigned to teach at that school, in order to do this, it was necessary that he get a masters degree. Wenninger was sent to the University of Ottawa, in Canada, there he studied symbolic logic under Thomas Greenwood of the philosophy department. His thesis title was “The Concept of Number According to Roger Bacon, after completing his degree, Wenninger went to the school in the Bahamas, where he was asked by the headmaster to choose between teaching English or math. Wenninger chose math, since it seemed to be more in line with the topic of his MA thesis, however, not having taken many math courses in college, Wenninger admits to being able to teach by staying a few pages ahead of the students. He taught Algebra, Euclidean Geometry, Trigonometry and Analytic Geometry, after ten years of teaching, Wenninger felt he was becoming a bit stale. At the suggestion of his headmaster, Wenninger attended the Columbia Teachers College in summer sessions over a period in the late fifties. It was here that his interest in the “New Math” was formed, Wenninger died at the age of 97, at St Johns Abbey on Friday, February 17,2017. Wenninger’s first publication on the topic of polyhedra was the entitled, “Polyhedron Models for the Classroom”. He wrote to H. S. M. Coxeter and received a copy of Uniform polyhedra which had a complete list of all 75 uniform polyhedra, after this, he spent a great deal of time building various polyhedra. He made 65 of them and had them on display in his classroom, at this point, Wenninger decided to contact a publisher to see if there was any interest in a book. He had the models photographed and wrote the text, which he sent off to Cambridge University Press in London
5.
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
6.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
7.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
8.
Stellation diagram
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In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions, regions not intersected by any further lines are called elementary regions. Usually infinite regions are excluded from the diagram, along with any infinite portions of the lines, each elementary region represents a top face of one cell, and a bottom face of another. A collection of diagrams, one for each face type, can be used to represent any stellation of the polyhedron. A stellation diagram exists for every face of a given polyhedron, in face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solids will have different stellation diagrams for different kinds of faces, list of Wenninger polyhedron models The fifty nine icosahedra M Wenninger, Polyhedron models, Cambridge University Press, 1st Edn, Ppbk. Coxeter, Harold Scott MacDonald, Du Val, P. Flather, H. T. Petrie, J. F
9.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations
10.
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
11.
Truncated icosahedron
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In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. It has 12 regular pentagonal faces,20 regular hexagonal faces,60 vertices and 90 edges and it is the Goldberg polyhedron GPV or 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs typically patterned with white hexagons, geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene C60 molecule and it is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb. This polyhedron can be constructed from an icosahedron with the 12 vertices truncated such that one third of each edge is cut off at each of both ends and this creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges, cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of, where φ =1 + √5/2 is the golden mean. Using φ2 = φ +1 one verifies that all vertices are on a sphere, centered at the origin, with the radius equal to √9φ +10. Permutations, X axis Y axis Z axis The truncated icosahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane and this result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge is approximately 23. 281446°. The area A and the volume V of the truncated icosahedron of edge length a are, with unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron easily demonstrates the Euler characteristic,32 +60 −90 =2, the balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more due to the pressure of the air inside. This ball type was introduced to the World Cup in 1970, geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the wheels used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix. This shape was also the configuration of the used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs. The truncated icosahedron can also be described as a model of the Buckminsterfullerene, or buckyball, molecule, an allotrope of elemental carbon, discovered in 1985
12.
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
13.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
14.
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
15.
Regular polyhedron
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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive, in classical contexts, many different equivalent definitions are used, a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form, there are 5 finite convex regular polyhedra, known as the Platonic solids. These are the, tetrahedron, cube, octahedron, dodecahedron and icosahedron, there are also four regular star polyhedra, making nine regular polyhedra in all. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons, All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre, An insphere, tangent to all faces, an intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices, the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them, Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry, the five Platonic solids have an Euler characteristic of 2. Some of the stars have a different value. The sum of the distances from any point in the interior of a polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra, in a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows, The tetrahedron is self-dual, the cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other, the small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other, the Schläfli symbol of the dual is just the original written backwards, for example the dual of is. See also Regular polytope, History of discovery, stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery, the earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, euclids reference to Plato led to their common description as the Platonic solids
16.
Enneagram (geometry)
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In geometry, an enneagram is a nine-pointed plane figure. It is sometimes called a nonagram, the name enneagram combines the numeral prefix, ennea-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς meaning a line, a regular enneagram is constructed using the same points as the regular enneagon but connected in fixed steps. It has two forms, represented by a Schläfli symbol as and, connecting every second and every fourth points respectively, there is also a star figure, or 3, made from the regular enneagon points but connected as a compound of three equilateral triangles. This star figure is known as the star of Goliath, after or 2. The nine-pointed star or enneagram can also symbolize the nine gifts or fruits of the Holy Spirit, the heavy metal band Slipknot uses the star figure enneagram as a symbol. Nonagon List of regular star polygons Bibliography John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Nonagram -- from Wolfram MathWorld
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Polyhedron model
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A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. Polyhedron models are found in mathematics classrooms much as globes in geography classrooms, Polyhedron models are notable as three-dimensional proof-of-concepts of geometric theories. Some polyhedra also make great centerpieces, tree toppers, Holiday decorations, the Merkaba religious symbol, for example, is a stellated octahedron. Constructing large models offer challenges in engineering structural design, construction begins by choosing a size of the model, either the length of its edges or the height of the model. The size will dictate the material, the adhesive for edges, the construction time, a single-colour cardboard model is easiest to construct — and some models can be made by folding a pattern, called a net, from a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron, one way is to colour each face differently. A second way is to all square faces the same, all pentagonal faces the same. A third way is to colour opposite faces the same, many polyhedra are also coloured such that no same-coloured faces touch each other along an edge or at a vertex. For example, a 20-face icosahedron can use twenty colours, one colour, ten colours, or five colours, an alternative way for polyhedral compound models is to use a different colour for each polyhedron component. One way is to copy templates from a book, such as Magnus Wenningers Polyhedron Models,1974. A second way is drawing faces on paper or with computer-aided design software, the exposed nets of the faces are then traced or printed on template material. A third way is using the software named Stella to print nets, a model, particularly a large one, may require another polyhedron as its inner structure or as a construction mold. A suitable inner structure prevents the model from collapsing from age or stress, the net templates are then replicated onto the material, matching carefully the chosen colours. Cardboard nets are usually cut with tabs on each edge, so the step for cardboard nets is to score each fold with a knife. Panelboard nets, on the hand, require molds and cement adhesives. Assembling multi-colour models is easier with a model of a simpler related polyhedron used as a colour guide, complex models, such as stellations, can have hundreds of polygons in their nets. Recent computer graphics technologies allow people to rotate 3D polyhedron models on a video screen in all three dimensions. Recent technologies even provide shadows and textures for a realistic effect
18.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
19.
Harmonices Mundi
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Harmonices Mundi is a book by Johannes Kepler. In the work Kepler discusses harmony and congruence in geometrical forms, the final section of the work relates his discovery of the so-called third law of planetary motion. Thus Kepler could reason that his relationships gave evidence for God acting as a geometer, rather than a Pythagorean numerologist. The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler. Musica Universalis was a philosophical metaphor that was taught in the quadrivium. He notes musical harmony as being a product of man, derived from angles, in turn, this allowed Kepler to claim the Earth has a soul because it is subjected to astrological harmony. Chapters 1 and 2 of The Harmony of the World contain most of Kepler’s contributions concerning polyhedra and he is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather and he returns to this concept later in Harmonices Mundi with relation to astronomical explanations. He describes polyhedra in terms of their faces, which is similar to the used in Plato’s Timaeus to describe the formation of Platonic solids in terms of basic triangles. While medieval philosophers spoke metaphorically of the music of the spheres and he found that the difference between the maximum and minimum angular speeds of a planet in its orbit approximates a harmonic proportion. For instance, the angular speed of the Earth as measured from the Sun varies by a semitone, from mi to fa. Venus only varies by a tiny 25,24 interval, Kepler explains the reason for the Earths small harmonic range, The Earth sings Mi, Fa, Mi, you may infer even from the syllables that in this our home misery and famine hold sway. The celestial choir Kepler formed was made up of a tenor, at very rare intervals all of the planets would sing together in perfect concord, Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring orbits approximate musical harmonies within a margin of error of less than a diesis. The orbits of Mars and Jupiter produce the one exception to this rule, in fact, the cause of Keplers dissonance might be explained by the fact that the asteroid belt separates those two planetary orbits, as discovered in 1801,150 years after Keplers death. Keplers previous book Astronomia nova related the discovery of the first two of the principles that we know today as Keplers laws, a copy of the 1619 edition was stolen from the National Library of Sweden in the 1990s. A small number of recent compositions either make reference to or are based on the concepts of Harmonices Mundi or Harmony of the Spheres, the most notable of these are, Laurie Spiegel, Keplers Harmony of the Worlds. An excerpt of the piece was selected by Carl Sagan for inclusion on the Voyager Golden Record, mike Oldfield, Music of the Spheres
20.
Small stellated dodecahedron
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In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol. It is one of four regular polyhedra. It is composed of 12 pentagrammic faces, with five meeting at each vertex. It shares the vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron and it is the second of four stellations of the dodecahedron. It is central to two lithographs by M. C and its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron, compound of small stellated dodecahedron and great dodecahedron Small stellated dodecahedron programing Wenninger, Magnus. Weber, Matthias, Keplers small stellated dodecahedron as a Riemann surface,220, 167–182 Eric W. Weisstein, Small stellated dodecahedron at MathWorld
21.
Great stellated dodecahedron
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In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol. It is one of four regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three meeting at each vertex. It shares its vertex arrangement with the regular dodecahedron, as well as being a stellation of a dodecahedron and it is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the icosahedron, is related in a similar fashion to the icosahedron. Shaving the triangular pyramids off results in an icosahedron, if the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra, truncating edges down to points produces the great icosidodecahedron as a rectified great stellated dodecahedron. The process completes as a birectification, reducing the original faces down to points, eric W. Weisstein, Great stellated dodecahedron at MathWorld
22.
Louis Poinsot
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Louis Poinsot was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of acting on a rigid body could be resolved into a single force. —Louis Poinsot, Théorie nouvelle de la rotation des corps Louis was born in Paris on 3 January 1777 and he attended the school of Lycée Louis-le-Grand for secondary preparatory education for entrance to the famous École Polytechnique. In October 1794, at age 17, he took the École Polytechnique entrance exam, a student there for two years, he left in 1797 to study at École des Ponts et Chaussées to become a civil engineer. Although now on course for the practical and secure professional study of engineering, he discovered his true passion. Poinsot thus left the École des Ponts et Chaussées and civil engineering to become a teacher at the secondary school Lycée Bonaparte in Paris. From there he became general of the Imperial University of France. He shared the post with another famous mathematician, Delambre, on 1 November 1809, Poinsot became assistant professor of analysis and mechanics at his old school the École Polytechnique. He also worked at the famous Bureau des Longitudes from 1839 until his death, on the death of Joseph-Louis Lagrange in 1813, Poinsot was elected to fill his place at the Académie des Sciences. In 1840 he became a member of the council of public instruction. In 1846 he was awarded an Officer of the Legion of Honor, Poinsot was elected Fellow of the Royal Society of London in 1858. He died in Paris on 5 December 1859, from the diary of Thomas Hirst,20 December 1857. Shook me kindly by the hand, bid me be seated and he is now between 60 and 70 years old, with silver silken hair neatly arranged on a fine intelligent head. He is tall and thin, but although he now stoops with age and he was loosely but neatly dressed in a large ample robe de chambre. His features are finely moulded — indeed everything about the man betokens good blood, I did not misunderstand a word, although he spoke always in a low tone, and now and then his voice dropped as if from weariness, but he never wandered from his point. The crater Poinsot on the moon is named after Poinsot, a street in Paris is called Rue Poinsot. Gustave Eiffel included Poinsot among the 72 names of prominent French scientists on plaques around the first stage of the Eiffel Tower, Poinsot was determined to publish only fully developed results and to present them with clarity and elegance. Consequently he left a rather limited body of work, in particular he devised, what is now known as, Poinsots construction
23.
Great icosahedron
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In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol and Coxeter-Dynkin diagram of. It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence, the great icosahedron can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry. This construction can be called a tetrahedron or retrosnub tetratetrahedron, similar to the snub tetrahedron symmetry of the icosahedron. It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or and it shares the same vertex arrangement as the regular convex icosahedron. It also shares the same arrangement as the small stellated dodecahedron. A truncation operation, repeatedly applied to the icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a great icosahedron. The process completes as a birectification, reducing the original faces down to points, coxeter, Harold Scott MacDonald, Du Val, P. Flather, H. T. Petrie, J. F. Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8,3.66.2 Stellating the Platonic solids, pp. 96-104 Eric W. Weisstein, Great icosahedron at MathWorld
24.
Great dodecahedron
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In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of. It is one of four regular polyhedra. It is composed of 12 pentagonal faces, with five meeting at each vertex. It shares the same arrangement as the convex regular icosahedron. If the great dodecahedron is considered as a properly intersected surface geometry, the excavated dodecahedron can be seen as the same process applied to a regular dodecahedron. A truncation process applied to the great dodecahedron produces a series of uniform polyhedra. Truncating edges down to points produces the dodecadodecahedron as a great dodecahedron. The process completes as a birectification, reducing the original faces down to points and this shape was the basis for the Rubiks Cube-like Alexanders Star puzzle. The great dodecahedron provides an easy mnemonic for the binary Golay code Compound of small stellated dodecahedron and great dodecahedron Eric W. Weisstein, uniform polyhedra and duals Metal sculpture of Great Dodecahedron
25.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
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Harold Scott MacDonald 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
27.
Magnus Wenninger
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Father Magnus J. Wenninger OSB was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Born to German immigrants in Park Falls, Wisconsin, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, the ad was for a preparatory school in Collegeville, Minnesota, associated with the Benedictine St. John’s University. While admitting to feeling homesick at first, Wenninger quickly made friends and, after a year, knew that this was where he needed to be. He was a student in a section of the school that functioned as a “minor seminary” – later moving on into St. John’s where he studied philosophy and theology. Wenninger became a Benedictine monk, he took on his monastic name Magnus, at the start of his career, Wenninger did not set out on a path one might expect would lead to his becoming the great polyhedronist that he is known as today. Rather, a few chance happenings and seemingly minor decisions shaped a course for Wenninger that led to his groundbreaking studies, shortly after becoming a priest, Wenninger’s Abbot informed him that their order was starting up a school in the Bahamas. It was decided that Wenninger would be assigned to teach at that school, in order to do this, it was necessary that he get a masters degree. Wenninger was sent to the University of Ottawa, in Canada, there he studied symbolic logic under Thomas Greenwood of the philosophy department. His thesis title was “The Concept of Number According to Roger Bacon, after completing his degree, Wenninger went to the school in the Bahamas, where he was asked by the headmaster to choose between teaching English or math. Wenninger chose math, since it seemed to be more in line with the topic of his MA thesis, however, not having taken many math courses in college, Wenninger admits to being able to teach by staying a few pages ahead of the students. He taught Algebra, Euclidean Geometry, Trigonometry and Analytic Geometry, after ten years of teaching, Wenninger felt he was becoming a bit stale. At the suggestion of his headmaster, Wenninger attended the Columbia Teachers College in summer sessions over a period in the late fifties. It was here that his interest in the “New Math” was formed, Wenninger died at the age of 97, at St Johns Abbey on Friday, February 17,2017. Wenninger’s first publication on the topic of polyhedra was the entitled, “Polyhedron Models for the Classroom”. He wrote to H. S. M. Coxeter and received a copy of Uniform polyhedra which had a complete list of all 75 uniform polyhedra, after this, he spent a great deal of time building various polyhedra. He made 65 of them and had them on display in his classroom, at this point, Wenninger decided to contact a publisher to see if there was any interest in a book. He had the models photographed and wrote the text, which he sent off to Cambridge University Press in London
28.
Echidna
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Echidnas /ᵻˈkɪdnə/, sometimes known as spiny anteaters, belong to the family Tachyglossidae in the monotreme order of egg-laying mammals. The four extant species, together with the platypus, are the surviving members of the order Monotremata and are the only living mammals that lay eggs. The diet of species consists of ants and termites. Echidnas live in Australia and New Guinea, echidnas evidently evolved between 20 and 50 million years ago, descending from a platypus-like monotreme. This ancestor was aquatic, but echidnas adapted to life on land, the echidnas are named after Echidna, a creature from Greek mythology who was half-woman, half-snake, as the animal was perceived to have qualities of both mammals and reptiles. Echidnas are medium-sized, solitary mammals covered with hair and spines. Superficially, they resemble the anteaters of South America and other mammals such as hedgehogs. They are usually black or brown in colour, there have been several reports of albino echidnas, their eyes pink and their spines white. They have elongated and slender snouts that function as mouth and nose. They have very short, strong limbs with large claws, and are powerful diggers, echidnas have tiny mouths and toothless jaws. The echidna feeds by tearing open soft logs, anthills and the like, and using its long, sticky tongue, the echidnas ears are slits on the sides of their heads that are usually unseen due to the fact that they are blanketed by their spines. The external ear is created by a large funnel, deep in the muscle. The short-beaked echidnas diet consists largely of ants and termites, while the Zaglossus species typically eat worms, the tongues of long-beaked echidnas have sharp, tiny spines that help them capture their prey. They have no teeth, and break down their food by grinding it between the bottoms of their mouths and their tongues, echidnas faeces are 7 centimetres long and are cylindrical in shape, they are usually broken and unrounded, and composed largely of dirt and ant-hill material. Echidnas do not tolerate extreme temperatures, they use caves and rock crevices to shelter from weather conditions. Echidnas are found in forests and woodlands, hiding under vegetation and they sometimes use the burrows of animals such as rabbits and wombats. Individual echidnas have large, mutually overlapping territories, when swimming, they expose their snout and some of their spines. They are known to journey to water in order to groom, echidnas and the platypus are the only egg-laying mammals, known as monotremes
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Mammal
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Mammals are any vertebrates within the class Mammalia, a clade of endothermic amniotes distinguished from reptiles by the possession of a neocortex, hair, three middle ear bones and mammary glands. All female mammals nurse their young with milk, secreted from the mammary glands, Mammals include the largest animals on the planet, the great whales. The basic body type is a quadruped, but some mammals are adapted for life at sea, in the air, in trees. The largest group of mammals, the placentals, have a placenta, Mammals range in size from the 30–40 mm bumblebee bat to the 30-meter blue whale. With the exception of the five species of monotreme, all modern mammals give birth to live young, most mammals, including the six most species-rich orders, belong to the placental group. The largest orders are the rodents, bats and Soricomorpha, the next three biggest orders, depending on the biological classification scheme used, are the Primates, the Cetartiodactyla, and the Carnivora. Living mammals are divided into the Yinotheria and Theriiformes There are around 5450 species of mammal, in some classifications, extant mammals are divided into two subclasses, the Prototheria, that is, the order Monotremata, and the Theria, or the infraclasses Metatheria and Eutheria. The marsupials constitute the group of the Metatheria, and include all living metatherians as well as many extinct ones. Much of the changes reflect the advances of cladistic analysis and molecular genetics, findings from molecular genetics, for example, have prompted adopting new groups, such as the Afrotheria, and abandoning traditional groups, such as the Insectivora. The mammals represent the only living Synapsida, which together with the Sauropsida form the Amniota clade, the early synapsid mammalian ancestors were sphenacodont pelycosaurs, a group that produced the non-mammalian Dimetrodon. At the end of the Carboniferous period, this group diverged from the line that led to todays reptiles. Some mammals are intelligent, with some possessing large brains, self-awareness, Mammals can communicate and vocalize in several different ways, including the production of ultrasound, scent-marking, alarm signals, singing, and echolocation. Mammals can organize themselves into fission-fusion societies, harems, and hierarchies, most mammals are polygynous, but some can be monogamous or polyandrous. They provided, and continue to provide, power for transport and agriculture, as well as commodities such as meat, dairy products, wool. Mammals are hunted or raced for sport, and are used as model organisms in science, Mammals have been depicted in art since Palaeolithic times, and appear in literature, film, mythology, and religion. Defaunation of mammals is primarily driven by anthropogenic factors, such as poaching and habitat destruction, Mammal classification has been through several iterations since Carl Linnaeus initially defined the class. No classification system is accepted, McKenna & Bell and Wilson & Reader provide useful recent compendiums. Though field work gradually made Simpsons classification outdated, it remains the closest thing to a classification of mammals
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Spine (zoology)
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In a zoological context, a spine is a hard, needle-like anatomical structure. Spines are found in a range of animals both vertebrate and invertebrate. In most spiny mammals, the spines are modified hairs, with a spongy center covered in a thick, hard layer of keratin, spines in mammals include the prickles of hedgehogs and the quills of porcupines as well as the prickly fur of spiny mice and Tenrec. They are also found on Echidna, a monotreme, the ancient precursor to mammals, Dimetrodon, had extremely long spines on their backbone that were joined together with a web of skin to form a sail-like structure. Many mammalian species also have penile spines, the Mesozoic eutriconodont mammal Spinolestes already displayed spines similar to those of modern spiny mice. Spines are found in the rays of certain finned bony fishes including scorpion fish, the sting that is found in a stingray is a type of barbed spine. Spines are also found in animals, such as sea urchins. They are a feature of the shell of a number of different species of gastropod and bivalve mollusks, spines are also found in internal organs in invertebrates, such as the copulatory spines in the male or female organs of certain flatworms. In many cases, spines are a mechanism that help protect the animal against potential predators. Because spines are sharp, they can puncture skin and inflict pain, the spine of some animals are capable of injecting venom. In the case of large species of stingray, a puncture with the barbed spine. Animals such as porcupines are considered aposematic, because their spines warn predators that they are dangerous, porcupines rattle their quills as a warning to predators, much like rattlesnakes. Because many species of fish and invertebrates carry venom within their spines, venom can cause intense pain, and can sometimes result in death if left untreated. On the other hand, being pricked by a porcupine quill is not dangerous, the quill can be removed by gently but firmly pulling it out of the skin. The barbed tip sometimes breaks off, but it works its way out through the skin over time. e, through the nose Pens Some of the earliest pens were made from quills Sometimes, Brush
31.
Polytope model
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The polyhedral model is a mathematical framework for loop nest optimization in program optimization. The following C code implements a form of error-distribution dithering similar to Floyd–Steinberg dithering, the two-dimensional array src contains h rows of w pixels, each pixel having a grayscale value between 0 and 255 inclusive. After the routine has finished, the output array dst will contain only pixels with value 0 or value 255, during the computation, each pixels dithering error is collected by adding it back into the src array. Each iteration of the inner loop modifies the values in src based on the values of src, src and we can illustrate the dependencies of src graphically, as in the diagram on the right. Performing the affine transformation = on the original dependency diagram gives us a new diagram and we can then rewrite the code to loop on p and t instead of i and j, obtaining the following skewed routine. A Practical Automatic Polyhedral Parallelizer and Locality Optimizer, proceedings of the 29th ACM SIGPLAN Conference on Programming Language Design and Implementation. New York, NY, USA, ACM, 101–113, Polyhedral. info - A website that collect information about polyhedral compilation Polly - LLVM Framework for High-Level Loop and Data-Locality Optimizations
32.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
33.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
34.
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
35.
Star polygon
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In geometry, a star polygon is a type of non-convex polygon. Only the regular polygons have been studied in any depth. The first usage is included in polygrams which includes polygons like the pentagram, star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram. The prefix is normally a Greek cardinal, but synonyms using other prefixes exist, for example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή meaning a line, alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. A regular star polygon is denoted by its Schläfli symbol, where p and q are relatively prime, the symmetry group of is dihedral group Dn of order 2n, independent of k. A regular star polygon can also be obtained as a sequence of stellations of a regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine, if p and q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example will appear as a triangle, but can be labeled with two sets of vertices 1-6 and this should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon. For |n/d|, the vertices have an exterior angle, β. These polygons are often seen in tiling patterns, the parametric angle α can be chosen to match internal angles of neighboring polygons in a tessellation pattern. The interior of a polygon may be treated in different ways. Three such treatments are illustrated for a pentagram, branko Grunbaum and Geoffrey Shephard consider two of them, as regular star polygons and concave isogonal 2n-gons. These include, Where a side occurs, one side is treated as outside and this is shown in the left hand illustration and commonly occurs in computer vector graphics rendering. The number of times that the polygonal curve winds around a given region determines its density, the exterior is given a density of 0, and any region of density >0 is treated as internal. This is shown in the illustration and commonly occurs in the mathematical treatment of polyhedra. Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure and this is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer, star polygons feature prominently in art and culture
36.
Wayback Machine
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The Internet Archive launched the Wayback Machine in October 2001. It was set up by Brewster Kahle and Bruce Gilliat, and is maintained with content from Alexa Internet, the service enables users to see archived versions of web pages across time, which the archive calls a three dimensional index. Since 1996, the Wayback Machine has been archiving cached pages of websites onto its large cluster of Linux nodes and it revisits sites every few weeks or months and archives a new version. Sites can also be captured on the fly by visitors who enter the sites URL into a search box, the intent is to capture and archive content that otherwise would be lost whenever a site is changed or closed down. The overall vision of the machines creators is to archive the entire Internet, the name Wayback Machine was chosen as a reference to the WABAC machine, a time-traveling device used by the characters Mr. Peabody and Sherman in The Rocky and Bullwinkle Show, an animated cartoon. These crawlers also respect the robots exclusion standard for websites whose owners opt for them not to appear in search results or be cached, to overcome inconsistencies in partially cached websites, Archive-It. Information had been kept on digital tape for five years, with Kahle occasionally allowing researchers, when the archive reached its fifth anniversary, it was unveiled and opened to the public in a ceremony at the University of California, Berkeley. Snapshots usually become more than six months after they are archived or, in some cases, even later. The frequency of snapshots is variable, so not all tracked website updates are recorded, Sometimes there are intervals of several weeks or years between snapshots. After August 2008 sites had to be listed on the Open Directory in order to be included. As of 2009, the Wayback Machine contained approximately three petabytes of data and was growing at a rate of 100 terabytes each month, the growth rate reported in 2003 was 12 terabytes/month, the data is stored on PetaBox rack systems manufactured by Capricorn Technologies. In 2009, the Internet Archive migrated its customized storage architecture to Sun Open Storage, in 2011 a new, improved version of the Wayback Machine, with an updated interface and fresher index of archived content, was made available for public testing. The index driving the classic Wayback Machine only has a bit of material past 2008. In January 2013, the company announced a ground-breaking milestone of 240 billion URLs, in October 2013, the company announced the Save a Page feature which allows any Internet user to archive the contents of a URL. This became a threat of abuse by the service for hosting malicious binaries, as of December 2014, the Wayback Machine contained almost nine petabytes of data and was growing at a rate of about 20 terabytes each week. Between October 2013 and March 2015 the websites global Alexa rank changed from 162 to 208, in a 2009 case, Netbula, LLC v. Chordiant Software Inc. defendant Chordiant filed a motion to compel Netbula to disable the robots. Netbula objected to the motion on the ground that defendants were asking to alter Netbulas website, in an October 2004 case, Telewizja Polska USA, Inc. v. Echostar Satellite, No.02 C3293,65 Fed. 673, a litigant attempted to use the Wayback Machine archives as a source of admissible evidence, Telewizja Polska is the provider of TVP Polonia and EchoStar operates the Dish Network
37.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
38.
Regular Polytopes (book)
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Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973, the book is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter twenty-four years to complete, regular Polytopes is a standard reference work on regular polygons, polyhedra and their higher dimensional analogues. It is unusual in the breadth of its coverage, its combination of mathematical rigour with geometric insight, Coxeter starts by introducing two-dimensional polygons and three-dimensional polyhedra. He then gives a rigorous definition of regularity and uses it to show that there are no other convex regular polyhedra apart from the five Platonic solids. The concept of regularity is extended to non-convex shapes such as star polygons and star polyhedra, to tessellations and honeycombs, Coxeter introduces and uses the groups generated by reflections that became known as Coxeter groups. The book combines algebraic rigour with clear explanations, many of which are illustrated with diagrams, the black and white plates in the book show solid models of three-dimensional polyhedra, and wire-frame models of projections of some higher-dimensional polytopes. At the end of each chapter Coxeter includes an Historical remarks section which provides a perspective of the development of the subject. The first two have been expounded by Sommerville and Neville, and we shall presuppose some familiarity with such treatises. Concerning the third, Poincaré wrote, A man who pursues it, will end up holding on to the fourth dimension. ”The original 1948 edition received a more complete review by M. Goldberg in MR0027148