1.
Dodecagon
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In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=
2.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
3.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
4.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
5.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
6.
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
7.
Polytope
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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli, the German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a class of objects. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes and they represent different approaches to generalizing the convex polytopes to include other objects with similar properties. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold, an example of this approach defines a polytope as a set of points that admits a simplicial decomposition. However this definition does not allow star polytopes with interior structures, the discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra and this approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms polytope and polyhedron are used in a different sense and this terminology is typically confined to polytopes and polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells, terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically, authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element, the terms adopted in this article are given in the table below, An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope, Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope and these bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point, a 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, the convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite, Polytopes are defined in this way, e. g. in linear programming
8.
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
9.
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
10.
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
11.
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
12.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
13.
Stella octangula
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The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula, a given to it by Johannes Kepler in 1609. It was depicted in Paciolis Divina Proportione,1509 and it is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It can also be seen as one of the stages in the construction of a 3D Koch Snowflake, the stellated octahedron can be constructed in several ways, It is a stellation of the regular octahedron, sharing the same face planes. It is also a regular compound, when constructed as the union of two regular tetrahedra. It can be obtained as an augmentation of the regular octahedron, in this construction it has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids. It is a facetting of the cube, sharing the vertex arrangement, a compound of two spherical tetrahedra can be constructed, as illustrated. The two tetrahedra of the view of the stellated octahedron are desmic, meaning that each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. One of these two crossings is visible in the octahedron, the other crossing occurs at a point at infinity of the projective space. The same twelve tetrahedron vertices also form the points of Reyes configuration, the stella octangula numbers are figurate numbers that count the number of balls that can be arranged into the shape of a stellated octahedron. They are 0,1,14,51,124,245,426,679,1016,1449,1990, the stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Eschers print Stars, and provides the form in Eschers Double Planetoid. Peter R. Cromwell, Polyhedra, Cambridge University Press Polyhedra H. S. M. Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8,3.6 The five regular compounds, pp. 47-50,6.2 Stellating the Platonic solids, pp. 96-104 Weisstein, Weisstein, Eric W. Compound of two tetrahedra
14.
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
15.
Co-prime
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
16.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
17.
Hexagram
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A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol,2, or. It is the compound of two equilateral triangles, the intersection is a regular hexagon. It is used in historical, religious and cultural contexts, for example in Hanafism, Jewish identity, in mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots. A six-pointed star, like a hexagon, can be created using a compass. Without changing the radius of the compass, set its pivot on the circles circumference, with the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles. It is possible that as a geometric shape, like for example the triangle, circle, or square. The hexagram is a symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes the nara-narayana, or perfect meditative state of balance achieved between Man and God, and if maintained, results in moksha, or nirvana, some researchers have theorized that the hexagram represents the astrological chart at the time of Davids birth or anointment as king. The hexagram is also known as the Kings Star in astrological circles, in antique papyri, pentagrams, together with stars and other signs, are frequently found on amulets bearing the Jewish names of God, and used to guard against fever and other diseases. Curiously the hexagram is not found among these signs, in the Greek Magical Papyri at Paris and London there are 22 signs side by side, and a circle with twelve signs, but neither a pentagram nor a hexagram. Six-pointed stars have also found in cosmological diagrams in Hinduism, Buddhism. The reasons behind this symbols common appearance in Indic religions and the West are unknown, one possibility is that they have a common origin. The other possibility is that artists and religious people from several cultures independently created the hexagram shape, within Indic lore, the shape is generally understood to consist of two triangles—one pointed up and the other down—locked in harmonious embrace. The two components are called Om and the Hrim in Sanskrit, and symbolize mans position between earth and sky, the downward triangle symbolizes Shakti, the sacred embodiment of femininity, and the upward triangle symbolizes Shiva, or Agni Tattva, representing the focused aspects of masculinity. The mystical union of the two triangles represents Creation, occurring through the union of male and female. The two locked triangles are known as Shanmukha—the six-faced, representing the six faces of Shiva & Shaktis progeny Kartikeya. This symbol is also a part of several yantras and has significance in Hindu ritual worship
18.
Pentagram
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A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον, from πέντε, five + γραμμή, the word pentacle is sometimes used synonymously with pentagram The word pentalpha is a learned modern revival of a post-classical Greek name of the shape. The pentagram is the simplest regular star polygon, the pentagram contains ten points and fifteen line segments. It is represented by the Schläfli symbol, like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10. The pentagram can be constructed by connecting alternate vertices of a pentagon and it can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Each intersection of edges sections the edges in the golden ratio, also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges is φ. As the four-color illustration shows, r e d g r e e n = g r e e n b l u e = b l u e m a g e n t a = φ. The pentagram includes ten isosceles triangles, five acute and five obtuse isosceles triangles, in all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles, the obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. The pentagram of Venus is the apparent path of the planet Venus as observed from Earth, the tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram. Groups of five intersections of curves, equidistant from the center, have the same geometric relationship. In early monumental Sumerian script, or cuneiform, a pentagram glyph served as a logogram for the word ub, meaning corner, angle, nook, the word Pentemychos was the title of the cosmogony of Pherecydes of Syros. Here, the five corners are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear. The pentangle plays an important symbolic role in the 14th-century English poem Sir Gawain, heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. By the mid-19th century a distinction had developed amongst occultists regarding the pentagrams orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, however, the influential writer Eliphas Levi called it evil whenever the symbol appeared the other way up. It is the goat of lust attacking the heavens with its horns and it is the sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns, faust, The pentagram thy peace doth mar
19.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
20.
Enneagon
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In geometry, a nonagon /ˈnɒnəɡɒn/ is a nine-sided polygon or 9-gon. The name nonagon is a hybrid formation, from Latin, used equivalently, attested already in the 16th century in French nonogone. The name enneagon comes from Greek enneagonon, and is more correct. A regular nonagon is represented by Schläfli symbol and has angles of 140°. Although a regular nonagon is not constructible with compass and straightedge and it can be also constructed using neusis, or by allowing the use of an angle trisector. The following is a construction of a nonagon using a straightedge. The regular enneagon has Dih9 symmetry, order 18, there are 2 subgroup dihedral symmetries, Dih3 and Dih1, and 3 cyclic group symmetries, Z9, Z3, and Z1. These 6 symmetries can be seen in 6 distinct symmetries on the enneagon, john Conway labels these by a letter and group order. Full symmetry of the form is r18 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g9 subgroup has no degrees of freedom but can seen as directed edges. The regular enneagon can tessellate the euclidean tiling with gaps and these gaps can be filled with regular hexagons and isosceles triangles. In the notation of symmetrohedron this tiling is called H with H representing *632 hexagonal symmetry in the plane, the K9 complete graph is often drawn as a regular enneagon with all 36 edges connected. This graph also represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex and they Might Be Giants have a song entitled Nonagon on their childrens album Here Come the 123s. It refers to both an attendee at a party at which everybody in the party is a many-sided polygon, slipknots logo is also a version of a nonagon, being a nine-pointed star made of three triangles. King Gizzard & the Lizard Wizard have an album titled Nonagon Infinity, temples of the Bahai Faith are required to be nonagonal. The U. S. Steel Tower is an irregular nonagon, enneagram Trisection of the angle 60°, Proximity construction Weisstein, Eric W. Nonagon
21.
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
22.
Heptagon
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In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon
23.
Heptagram
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A heptagram, septagram, or septogram is a seven-point star drawn with seven straight strokes. The name heptagram combines a numeral prefix, hepta-, with the Greek suffix -gram, the -gram suffix derives from γραμμῆς meaning a line. In general, a heptagram is any self-intersecting heptagon, there are two regular heptagrams, labeled as and, with the second number representing the vertex interval step from a regular heptagon. This is the smallest star polygon that can be drawn in two forms, as irreducible fractions, the two heptagrams are sometimes called the heptagram and the great heptagram. The previous one, the hexagram, is a compound of two triangles. The smallest star polygon is the pentagram, the next one is the octagram, followed by the regular enneagram, which also has two forms, and, as well as one compound of three triangles. The heptagram was used in Christianity to symbolize the seven days of creation, the heptagram is a symbol of perfection in many Christian sects. The heptagram is used in the symbol for Babalon in Thelema, the heptagram is known among neopagans as the Elven Star or Fairy Star. It is treated as a symbol in various modern pagan. Blue Star Wicca also uses the symbol, where it is referred to as a septegram, the second heptagram is a symbol of magical power in some pagan spiritualities. The heptagram is used by members of the otherkin subculture as an identifier. In alchemy, a star can refer to the seven planets which were known to ancient alchemists. The seven-pointed star is incorporated into the flags of the bands of the Cherokee Nation. The Bennington flag, a historical American Flag, has thirteen seven-pointed stars along with the numerals 76 in the canton, the Flag of Jordan contains a seven-pointed star. The Flag of Australia employs five heptagrams and one pentagram to depict the Southern Cross constellation, some old versions of the coat of arms of Georgia including the Georgian Soviet Socialist Republic used the heptagram as an element. A seven-pointed star is used as the badge in many sheriffs departments, the seven-pointed star is used as the logo for the international Danish shipping company A. P. Moller–Maersk Group, sometimes known simply as Maersk. In George R. R. Martins novel series A Song of Ice and Fire, Star polygon Stellated polygons Two-dimensional regular polytopes Bibliography Grünbaum, B. and G. C. Shephard, Tilings and Patterns, New York, W. H. Freeman & Co, polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes
24.
Octagram
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In geometry, an octagram is an eight-angled star polygon. The name octagram combine a Greek numeral prefix, octa-, with the Greek suffix -gram, the -gram suffix derives from γραμμή meaning line. In general, an octagram is any self-intersecting octagon, the regular octagram is labeled by the Schläfli symbol, which means an 8-sided star, connected by every third point. These variations have a dihedral, Dih4, symmetry, The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE. Deeper truncations of the square can produce intermediate star polygon forms with equal spaced vertices. A truncated square is an octagon, t=, a quasitruncated square, inverted as, is an octagram, t=. The uniform star polyhedron stellated truncated hexahedron, t=t has octagram faces constructed from the cube in this way, there are two regular octagrammic star figures of the form, the first constructed as two squares =2, and second as four degenerate digons, =4. There are other isogonal and isotoxal compounds including rectangular and rhombic forms, an octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines, usage Rub el Hizb - Islamic character Star of Lakshmi - Indian character Auseklis – usage of regular octagram by Latvians Guñelve – representation of Venus in Mapuche iconography. Selburose – usage of regular octagram in Norwegian design Stars generally Star Stellated polygons Two-dimensional regular polytopes Grünbaum, shephard, Tilings and Patterns, New York, W. H. Freeman & Co. Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes, etc. ed T. Bisztriczky et al. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Octagram
25.
Square (geometry)
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
26.
H.S.M. Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
27.
Archimedean solids
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
28.
Cube
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Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the lungs and the bloodstream, the International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen, beryllium copper is a non-ferrous alloy used in springs, spring wire, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
29.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
30.
Rhombic dodecahedron
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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types and it is a Catalan solid, and the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron and its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly √2 times the length of the diagonal, so that the acute angles on each face measure arccos. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane and this polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic crystals, some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron, the rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent, but the chemical bonds lie on the remaining edges, the graph of the rhombic dodecahedron is nonhamiltonian. The last two correspond to the B2 and A2 Coxeter planes, the rhombic dodecahedron is a parallelohedron, a space-filling polyhedron. Other symmetry constructions of the dodecahedron are also space-filling. For example, with 4 square faces, and 60-degree rhombic faces and it be seen as a cuboctahedron with square pyramids augmented on the top and bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces and it has the same topology but different geometry. The rhombic faces in this form have the golden ratio, another topologically equivalent variation, sometimes called a trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites. It has 8 vertices adjusted in or out in sets of 4. Variations can be parametrized by, where b is determined from a for planar faces and this polyhedron is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry
31.
Triakis tetrahedron
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In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron and it can be seen as a tetrahedron with triangular pyramids added to each face, that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name, the length of the shorter edges is 3/5 that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area 5/3√11, a triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will be coplanar and this can be seen by adding the 6 edges of tetrahedron inside of a cube. This chiral figure is one of thirteen stellations allowed by Millers rules, the triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design
32.
Rhombic triacontahedron
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In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types and it is a Catalan solid, and the dual polyhedron of the icosidodecahedron. The ratio of the diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1 = tan−1. A rhombus so obtained is called a golden rhombus, being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids and it contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra, the plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance. Using one of the three golden rectangles drawn into the inscribed icosahedron we can easily deduce the distance between the center of the solid and the center of its rhombic face. The rhombic triacontahedron can be dissected into 20 golden rhombohedra,10 acute ones and 10 flat ones, danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron, the simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube. Roger von Oechs Ball of Whacks comes in the shape of a rhombic triacontahedron, the rhombic triacontahedron is used as the d30 thirty-sided die, sometimes useful in some roleplaying games or other places. The rhombic triacontahedron has three positions, two centered on vertices, and one mid-edge. Embedded in projection 10 are the fat rhombus and skinny rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling, the rhombic triacontahedron has over 227 stellations. This polyhedron is a part of a sequence of rhombic polyhedra, the cube can be seen as a rhombic hexahedron where the rhombi are also rectangles. The rhombic triacontahedron forms the hull of one projection of a 6-cube to 3 dimensions. Truncated rhombic triacontahedron Rhombille tiling Golden rhombus Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design
33.
List of Wenninger polyhedron models
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This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building and it contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex regular and quasiregular polyhedra. This list was written to honor this early work from Wenninger. Models listed here can be cited as Wenninger Model Number N, the polyhedra are grouped in 5 tables, Regular, Semiregular, regular star polyhedra, Stellations and compounds, and uniform star polyhedra. The four regular polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings. List of uniform polyhedra The fifty nine icosahedra Wenninger, Magnus, errata In Wenninger, the vertex figure for W90 is incorrectly shown as having parallel edges. Magnus J. Wenninger Software used to generate images in this article, Stella, Polyhedron Navigator Stella - Can create and print nets for all of Wenningers polyhedron models
34.
Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids
35.
Chirality (mathematics)
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In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral, in 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane, a chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ, the hand, the most familiar chiral object, a non-chiral figure is called achiral or amphichiral. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. A right shoe is different from a left shoe only for being mirror images of each other, in contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped tetrominoes of the video game Tetris also exhibit chirality. Individually they contain no mirror symmetry in the plane, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In three dimensions, every figure that possesses a plane of symmetry S1, an inversion center of symmetry S2. Note, however, that there are achiral figures lacking both plane and center of symmetry, an example is the figure F0 = which is invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry. The figure F1 = also is achiral as the origin is a center of symmetry, note also that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry and its symmetry group is a frieze group generated by a single glide reflection. A knot is called if it can be continuously deformed into its mirror image. For example the unknot and the knot are achiral, whereas the trefoil knot is chiral. Chirality in Metric Spaces, Symmetry, Culture and Science 21, pp. 27–36 Chiral Polyhedra by Eric W. Weisstein, when Topology Meets Chemistry by Erica Flapan. Chiral manifold at the Manifold Atlas
36.
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
37.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
38.
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
39.
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
40.
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
41.
Second stellation of icosahedron
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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
42.
Final stellation of the icosahedron
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This polyhedron is the seventeenth stellation of the icosahedron, and given as Wenninger model index 42. As a geometrical figure, it has two interpretations, described below, As an irregular polyhedron with 20 identical self-intersecting enneagrammic faces,90 edges,60 vertices. As a simple polyhedron with 180 triangular faces,270 edges and this interpretation is useful for polyhedron model building. Johannes Kepler researched stellations that create regular star polyhedra in 1619,1619, In Harmonices Mundi, Johannes Kepler first applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. 1809, Louis Poinsot rediscovered Keplers polyhedra and two more, the icosahedron and great dodecahedron as regular star polyhedra, now called the Kepler–Poinsot polyhedra. 1812, Augustin-Louis Cauchy made a further enumeration of star polyhedra,1900, Max Brückner extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the complete stellation. 1924, A. H. Wheeler in 1924 published a list of 20 stellation forms,1938, In their 1938 book The Fifty Nine Icosahedra, H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie stated a set of rules for the regular icosahedron. The complete stellation is referenced as the eighth in the book,1974, In Wenningers 1974 book Polyhedron Models, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W42. 1995, Andrew Hume named it in his Netlib polyhedral database as the echidnahedron, the Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symbol of the stellation is H, because it includes all cells in the stellation diagram up to. As a simple, visible surface polyhedron, the form of the final stellation is composed of 180 triangular faces. These join along 270 edges, which in turn meet at 92 vertices, the 92 vertices lie on the surfaces of three concentric spheres. The radii of spheres are in the ratio 32,12,12. When regarded as a solid object with edge lengths a, φa, φ2a and φ2a√2 the complete icosahedron has surface area S =120 a 2. The complete stellation can also be seen as a star polyhedron having 20 faces corresponding to the 20 faces of the underlying icosahedron. Each face is an irregular 9/4 star polygon, or enneagram, since three faces meet at each vertex it has 20 ×9 /3 =60 vertices and 20 ×9 /2 =90 edges. When regarded as an icosahedron, the complete stellation is a noble polyhedron
43.
Facetting
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In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a polyhedron may be created along face diagonals or internal space diagonals. A faceted polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra, faceting is the reciprocal or dual process to stellation. For every stellation of some convex polytope, there exists a dual faceting of the dual polytope. For example, a regular pentagon has one symmetry faceting, the pentagram, the regular icosahedron can be faceted into three regular Kepler–Poinsot polyhedra, small stellated dodecahedron, great dodecahedron, and great icosahedron. The regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedron, the uniform stars and compound of five cubes are constructed by face diagonals. The excavated dodecahedron is a facetting with star hexagon faces, faceting has not been studied as extensively as stellation. In 1619, Kepler described a regular compound of two tetrahedra which fits inside a cube, and which he called the Stella octangula and this seems to be the first known example of faceting. In 1858, Bertrand derived the regular star polyhedra by faceting the regular icosahedron and dodecahedron. In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, in 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For a given vertex, the shows all the possible edges. It is dual to the dual polyhedrons stellation diagram, which all the possible edges and vertices for some face plane of the original core. Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de lAcadémie des Sciences,46, Facetting the dodecahedron, Acta crystallographica A30, pp. 548–552. Inchbald, G. Facetting diagrams, The mathematical gazette,90, alan Holden, Shapes, Space, and Symmetry. Archived from the original on 4 February 2007
44.
Duality (mathematics)
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues theorem is self-dual in this sense under the standard duality in projective geometry, many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. From a category theory viewpoint, duality can also be seen as a functor and this functor assigns to each space its dual space, and the pullback construction assigns to each arrow f, V → W its dual f∗, W∗ → V∗. In the words of Michael Atiyah, Duality in mathematics is not a theorem, the following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S, to any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S, taking the complement has the following properties, Applying it twice gives back the original set, i. e. c = A. This is referred to by saying that the operation of taking the complement is an involution, an inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X, a subset U of X is closed if, because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of sets is open, so dually. The interior of a set is the largest open set contained in it, because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the cone construction. Given a set C of points in the plane R2, unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C. Instead, C ∗ ∗ is the smallest cone containing C which may be bigger than C. Therefore this duality is weaker than the one above, in that Applying the operation twice gives back a possibly bigger set, the other two properties carry over without change, It is still true that an inclusion C ⊆ D is turned into an inclusion in the opposite direction. Given two subsets C and D of the plane, C is contained in D ∗ if, a very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V*. Its elements are the k-linear maps φ, V → k, the three properties of the dual cone carry over to this type of duality by replacing subsets of R2 by vector space and inclusions of such subsets by linear maps. That is, Applying the operation of taking the dual vector space twice gives another vector space V**, there is always a map V → V**
45.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
46.
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
47.
Great grand stellated 120-cell
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In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and has the vertex arrangement as the regular convex 120-cell. It is one of four regular star polychora discovered by Ludwig Schläfli and it is named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids, and the only one containing all three modifiers in the name. The great grand stellated 120-cell is the final stellation of the 120-cell, indeed, the great grand stellated 120-cell is dual to the grand 600-cell, which could be taken as a 4D analogue of the great icosahedron, dual of the great stellated dodecahedron. H. S. M. Coxeter, Regular Polytopes, 3rd, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 4D uniform polytopes o3o3o5/2x - gogishi, Regular polychora Discussion on names Reguläre Polytope The Regular Star Polychora Zome Model of the Final Stellation of the 120-cell
48.
Regular 4-polytope
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, Schläfli discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes and he skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures. That excludes cells and vertex figures as, and, the six convex and ten star polytopes described are the only solutions to these constraints. There are four nonconvex Schläfli symbols that have cells and vertex figures, and pass the dihedral test. The regular convex 4-polytopes are the analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions. Five of them may be thought of as close analogs of the Platonic solids, There is one additional figure, the 24-cell, which has no close three-dimensional equivalent. Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and these are fitted together along their respective faces in a regular fashion. The following tables lists some properties of the six convex regular 4-polytopes, the symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group, John Conway advocates the names simplex, orthoplex, tesseract, octaplex or polyoctahedron, dodecaplex or polydodecahedron, and tetraplex or polytetrahedron. The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analog of Eulers polyhedral formula, the topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients. The following table shows some 2-dimensional projections of these 4-polytopes, various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are given below the Schläfli symbol. The Schläfli–Hess 4-polytopes are the set of 10 regular self-intersecting star polychora. They are named in honor of their discoverers, Ludwig Schläfli, each is represented by a Schläfli symbol in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra and their names given here were given by John Conway, extending Cayleys names for the Kepler–Poinsot polyhedra, along with stellated and great, he adds a grand modifier
49.
120-cell
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In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C120, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid, the boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the dodecahedron and has called a dodecaplex. Just as a dodecahedron can be built up as a model with 12 pentagons,3 around each vertex, there are 120 cells,720 pentagonal faces,1200 edges, and 600 vertices. There are 4 dodecahedra,6 pentagons, and 4 edges meeting at every vertex, there are 3 dodecahedra and 3 pentagons meeting every edge. The dual polytope of the 120-cell is the 600-cell, the vertex figure of the 120-cell is a tetrahedron. The dihedral angle of the 120-cell is 144° The 600 vertices of the 120-cell include all permutations of, the 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces, one can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use, a stereographic projection, and a structure of intertwining rings. The cell locations lend themselves to a hyperspherical description, pick an arbitrary cell and label it the North Pole. Twelve great circle meridians radiate out in 3 dimensions, converging at the 5th South Pole cell and this skeleton accounts for 50 of the 120 cells. Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, the cells labeled interstitial in the following table do not fall on meridian great circles. Layers 2,4,6 and 8 cells are located over the cells faces. Layers 3 and 7s cells are located directly over the pole cells vertices, layer 5s cells are located over the pole cells edges. The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring, although the outer rings spiral around the inner ring, they actually have no helical torsion. The spiraling is a result of the 3-sphere curvature, the inner ring and the five outer rings now form a six ring, 60-cell solid torus
50.
Arthur Cayley
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Arthur Cayley F. R. S. was a British mathematician. He helped found the modern British school of pure mathematics, as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German and he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial and he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of groups, they had meant permutation groups, cayleys theorem is named in honour of Cayley. Arthur Cayley was born in Richmond, London, England, on 16 August 1821 and his father, Henry Cayley, was a distant cousin of Sir George Cayley the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled in Saint Petersburg, Russia, as a merchant and his mother was Maria Antonia Doughty, daughter of William Doughty. According to some writers she was Russian, but her fathers name indicates an English origin and his brother was the linguist Charles Bagot Cayley. Arthur spent his first eight years in Saint Petersburg, in 1829 his parents were settled permanently at Blackheath, near London. Arthur was sent to a private school, at age 14 he was sent to Kings College School. The schools master observed indications of genius and advised the father to educate his son not for his own business, as he had intended. At the unusually early age of 17 Cayley began residence at Trinity College, Cambridge, the cause of the Analytical Society had now triumphed, and the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects that had been suggested by reading the Mécanique analytique of Lagrange, cayleys tutor at Cambridge was George Peacock and his private coach was William Hopkins. He finished his course by winning the place of Senior Wrangler. His next step was to take the M. A. degree and he continued to reside at Cambridge University for four years, during which time he took some pupils, but his main work was the preparation of 28 memoirs to the Mathematical Journal. Because of the tenure of his fellowship it was necessary to choose a profession, like De Morgan, Cayley chose law. He made a specialty of conveyancing and it was while he was a pupil at the bar examination that he went to Dublin to hear Hamiltons lectures on quaternions. During this period of his life, extending over fourteen years, at Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, and is the chair that had been occupied by Isaac Newton
51.
Kepler-Poinsot polyhedra
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In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular dodecahedron and icosahedron. These figures have pentagrams as faces or vertex figures, the small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, in all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the structure and are sometimes called false edges. Likewise where three lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show golden balls at the vertices. For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid, the visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, each edge would now be divided into three shorter edges, and the 20 false vertices would become true ones, so that we have a total of 32 vertices. The hidden inner pentagons are no part of the polyhedral surface. Now the Eulers formula holds,60 −90 +32 =2, however this polyhedron is no longer the one described by the Schläfli symbol, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside. Because of this, they are not necessarily equivalent to the sphere as Platonic solids are. Schläfli held that all polyhedra must have χ =2, and he rejected the small stellated dodecahedron and this view was never widely held. The Kepler–Poinsot polyhedra exist in pairs, Small stellated dodecahedron and great dodecahedron and Great stellated dodecahedron. The small stellated dodecahedron and great icosahedron share the same vertices and edges, the icosahedron and great dodecahedron also share the same vertices and edges. The three dodecahedra are all stellations of the convex dodecahedron, and the great icosahedron is a stellation of the regular convex icosahedron. The small stellated dodecahedron and the icosahedron are facettings of the convex dodecahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, most, if not all, of the Kepler-Poinsot polyhedra were known of in some form or other before Kepler
52.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
53.
Polychoron
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In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements, vertices, edges, faces, each face is shared by exactly two cells. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron, topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space, similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space, a 4-polytope is a closed four-dimensional figure. It comprises vertices, edges, faces and cells, a cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i. e. it is not a compound, the most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. 4-polytopes cannot be seen in space due to their extra dimension. Several techniques are used to help visualise them, Orthogonal projection Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes. Perspective projection Just as a 3D shape can be projected onto a flat sheet, sectioning Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut hypersurface in three dimensions. A sequence of sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce an animation of these cross sections. The topology of any given 4-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, like all polytopes, 4-polytopes may be classified based on properties like convexity and symmetry. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the shapes of the non-convex star polygons. A 4-polytope is regular if it is transitive on its flags and this means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron
54.
Hemipolyhedron
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In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These hemi faces lie parallel to the faces of some other symmetrical polyhedron, the prefix hemi is also used to refer to certain projective polyhedra, such as the hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry. Their Wythoff symbols are of the form p/ p/q | r and they are thus related to the cantellated polyhedra, which have similar Wythoff symbols. The 2r-gon faces pass through the center of the model, if represented as faces of spherical polyhedra, they cover an entire hemisphere, the p/ notation implies a face turning backwards around the vertex figure. The nine forms, listed with their Wythoff symbols and vertex configurations are, only the octahemioctahedron represents an orientable surface, the remaining hemipolyhedra have non-orientable or single-sided surfaces. Since the hemipolyhedra have faces passing through the center, the figures have corresponding vertices at infinity, properly. In Magnus Wenningers Dual Models, they are represented with intersecting prisms, in practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, however, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. There are 9 such duals, sharing only 5 distinct outward forms, the members of a given identical pair differ in their arrangements of true and false vertices. The outwards forms are, The hemipolyhedra occur in pairs as facetings of the polyhedra with four faces at a vertex. These quasiregular polyhedra have vertex configuration m. n. m. n and their edges, in addition to forming the m- and n-gonal faces, thus, the hemipolyhedra can be derived from the quasiregular polyhedra by discarding either the m-gons or n-gons and then inserting the hemi faces. Since the hemipolyhedra, like the quasiregular polyhedra, also have two types of faces alternating around each vertex, they are also considered to be quasiregular. Here m and n correspond to p/q above, and h corresponds to 2r above, there are also related Euclidean tilings based on quasiregular tilings, that use apeirogons as the equivalents of the diametral 2r-gons above. Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, uniform polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences, The Royal Society,246, 401–450, doi,10. 1098/rsta.1954
55.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty, Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I, in modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching, mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts
56.
Mosaic
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A mosaic is a piece of art or image made from the assemblage of small pieces of colored glass, stone, or other materials. It is often used in art or as interior decoration. Most mosaics are made of small, flat, roughly square, pieces of stone or glass of different colors, some, especially floor mosaics, are made of small rounded pieces of stone, and called pebble mosaics. Others are made of other materials, mosaics have a long history, starting in Mesopotamia in the 3rd millennium BC. Pebble mosaics were made in Tiryns in Mycenean Greece, mosaics with patterns and pictures became widespread in classical times, Early Christian basilicas from the 4th century onwards were decorated with wall and ceiling mosaics. Mosaic fell out of fashion in the Renaissance, though artists like Raphael continued to practise the old technique, Roman and Byzantine influence led Jews to decorate 5th and 6th century synagogues in the Middle East with floor mosaics. Mosaic was widely used on buildings and palaces in early Islamic art, including Islams first great religious building, the Dome of the Rock in Jerusalem. Mosaic went out of fashion in the Islamic world after the 8th century, modern mosaics are made by professional artists, street artists, and as a popular craft. Many materials other than stone and ceramic tesserae may be employed, including shells, glass. The earliest known examples of made of different materials were found at a temple building in Abra, Mesopotamia. They consist of pieces of colored stones, shells and ivory, excavations at Susa and Chogha Zanbil show evidence of the first glazed tiles, dating from around 1500 BC. However, mosaic patterns were not used until the times of Sassanid Empire, mythological subjects, or scenes of hunting or other pursuits of the wealthy, were popular as the centrepieces of a larger geometric design, with strongly emphasized borders. Pliny the Elder mentions the artist Sosus of Pergamon by name, describing his mosaics of the left on a floor after a feast. Both of these themes were widely copied, most recorded names of Roman mosaic workers are Greek, suggesting they dominated high quality work across the empire, no doubt most ordinary craftsmen were slaves. Splendid mosaic floors are found in Roman villas across North Africa, in such as Carthage. The tiny tesserae allowed very fine detail, and an approach to the illusionism of painting, often small panels called emblemata were inserted into walls or as the highlights of larger floor-mosaics in coarser work. The normal technique was opus tessellatum, using larger tesserae, which was laid on site, there was a distinct native Italian style using black on a white background, which was no doubt cheaper than fully coloured work. In Rome, Nero and his architects used mosaics to cover surfaces of walls and ceilings in the Domus Aurea, built 64 AD
57.
Paolo Uccello
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Paolo Uccello, born Paolo di Dono, was an Italian painter and mathematician who was notable for his pioneering work on visual perspective in art. In his book, Lives of the Artists, Giorgio Vasari wrote that Uccello was obsessed by his interest in perspective, while his contemporaries used perspective to narrate different or succeeding stories, Uccello used perspective to create a feeling of depth in his paintings. His best known works are the three representing the battle of San Romano, which were wrongly entitled the Battle of Sant Egidio of 1416 for a long period of time. Paolo worked in the Late Gothic tradition, emphasizing colour and pageantry rather than the classical realism that other artists were pioneering and his style is best described as idiosyncratic, and he left no school of followers. He has had influence on twentieth-century art and literary criticism. The sources for Paolo Uccello’s life are few, Giorgio Vasari’s biography, written 75 years after Paolo’s death, due to the lack of sources, even his date of birth is questionable. It is believed that Uccello was born in Pratovecchio in 1397, and his tax declarations for some years indicate that he was born in 1397 and his father, Dono di Paolo, was a barber-surgeon from Pratovecchio near Arezzo, his mother, Antonia, was a high-born Florentine. His nickname Uccello came from his fondness for painting birds, from 1412 until 1416 he was apprenticed to the famous sculptor Lorenzo Ghiberti. Ghiberti was the designer of the doors of the Florence Baptistery, ghibertis late-Gothic, narrative style and sculptural composition greatly influenced Paolo. It was also around this time that Paolo began his friendship with Donatello. In 1414, Uccello was admitted to the guild, Compagnia di San Luca. These featured a scene that might well have impressed itself in the mind of the young Uccello. According to Vasari, Uccello’s first painting was a Saint Anthony between the saints Cosmas and Damianus, a commission for the hospital of Lelmo, next, he painted two figures in the convent of Annalena. Shortly afterwards, he painted three frescoes with scenes from the life of Saint Francis above the door of the Santa Trinita church. For the Santa Maria Maggiore church, he painted a fresco of the Annunciation, in this fresco, he painted a large building with columns in perspective. According to Vasari, people found this to be a great and beautiful achievement because this was the first example of how lines could be used to demonstrate perspective. As a result, this became a model for artists who wished to craft illusions of space in order to enhance the realness of their paintings. Paolo painted the Lives of the Church Fathers in the cloisters of the church of San Miniato, Uccello was asked to paint a number of scenes of distempered animals for the house of the Medici
58.
Basilica of St Mark, Venice
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The Patriarchal Cathedral Basilica of Saint Mark, commonly known as Saint Marks Basilica, is the cathedral church of the Roman Catholic Archdiocese of Venice, northern Italy. It is the most famous of the churches and one of the best known examples of Italo-Byzantine architecture. It lies at the end of the Piazza San Marco, adjacent. For its opulent design, gold mosaics, and its status as a symbol of Venetian wealth and power. The church was burned in a rebellion in 976, when the populace locked Pietro IV Candiano inside to kill him, nothing certain is known of the form of these early churches. From perhaps 1063 the present basilica was constructed, the consecration is variously recorded as being in 1084-5,1093,1102 and 1117, probably reflecting a series of consecrations of different parts. In 1094 the supposed body of Saint Mark was rediscovered in a pillar by Vitale Faliero, the building also incorporates a low tower, believed by some to have been part of the original Doges Palace. The Pala dOro ordered from Constantinople was installed on the altar in 1105. The main features of the present structure were all in place by then, except for the narthex or porch, and the facade. — The basic structure of the building has not been much altered. Its decoration has changed greatly over time, though the impression of the interior with a dazzling display of gold ground mosaics on all ceilings. Gradually, the exterior brickwork became covered with marble cladding and carvings, some older than the building itself. The latest structural additions include the closing-off of the Baptistery and St Isidors Chapel, the carvings on the facade and the Sacristy. During the 13th century the emphasis of the churchs function seems to have changed from being the chapel of the Doge to that of a state church. The transfer of the see was ordered by Napoleon during his period of control of Venice, before this, Venices cathedral from 1451 was the much less grand San Pietro di Castello. The procurators, an important organ of the Republic of Venice, were in charge of administration, their seats were the Procuratie, all building and restoring works were directed by the protos, great architects such as Jacopo Sansovino and Baldassarre Longhena held the office. The doge himself appointed a group of clergy led by the primicerio. Procurators and protos still exist and perform the tasks for the Patriarchate. The exterior of the west facade of the basilica is divided in three registers, lower, upper, and domes, in the lower register of the façade, five round-arched portals, enveloped by polychrome marble columns, open into the narthex through bronze-fashioned doors
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Italian Renaissance
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The term Renaissance is in essence a modern one that came into currency in the 19th century, in the work of historians such as Jules Michelet and Jacob Burckhardt. The French word renaissance means Rebirth, and the era is best known for the renewed interest in the culture of classical antiquity after the period that Renaissance humanists labeled the Dark Ages. Though today perhaps best known for Italian Renaissance art and architecture, the period saw major achievements in literature, music, philosophy, Italy became the recognized European leader in all these areas by the late 15th century, and to varying degrees retained this lead until about 1600. This was despite a turbulent and generally disastrous period in Italian politics, the European Renaissance began in Tuscany, and centred in the city of Florence. It later spread to Venice, where the remains of ancient Greek culture were brought together, the Renaissance later had a significant effect on Rome, which was ornamented with some structures in the new allantico mode, then was largely rebuilt by humanist sixteenth-century popes. The Italian Renaissance peaked in the century as foreign invasions plunged the region into the turmoil of the Italian Wars. However, the ideas and ideals of the Renaissance endured and spread into the rest of Europe, setting off the Northern Renaissance, the Italian Renaissance is best known for its cultural achievements. Accounts of Renaissance literature usually begin with Petrarch and his friend, famous vernacular poets of the 15th century include the renaissance epic authors Luigi Pulci, Matteo Maria Boiardo, and Ludovico Ariosto. 15th century writers such as the poet Poliziano and the Platonist philosopher Marsilio Ficino made extensive translations from both Latin and Greek, the same is true for architecture, as practiced by Brunelleschi, Leon Battista Alberti, Andrea Palladio, and Bramante. Their works include Florence Cathedral, St. Peters Basilica in Rome, yet cultural contributions notwithstanding, some present-day historians also see the era as one of the beginning of economic regression for Italy. By the Late Middle Ages, Latium, the heartland of the Roman Empire. Rome was a city of ancient ruins, and the Papal States were loosely administered, and vulnerable to external interference such as that of France, and later Spain. The Papacy was affronted when the Avignon Papacy was created in southern France as a consequence of pressure from King Philip the Fair of France, in the south, Sicily had for some time been under foreign domination, by the Arabs and then the Normans. Sicily had prospered for 150 years during the Emirate of Sicily, in contrast Northern and Central Italy had become far more prosperous, and it has been calculated that the region was among the richest of Europe. The Crusades had built lasting trade links to the Levant, the main trade routes from the east passed through the Byzantine Empire or the Arab lands and onwards to the ports of Genoa, Pisa, and Venice. Luxury goods bought in the Levant, such as spices, dyes, moreover, the inland city-states profited from the rich agricultural land of the Po valley. From France, Germany, and the Low Countries, through the medium of the Champagne fairs, land and river trade routes brought goods such as wool, wheat, and precious metals into the region. The extensive trade that stretched from Egypt to the Baltic generated substantial surpluses that allowed significant investment in mining, thus, while northern Italy was not richer in resources than many other parts of Europe, the level of development, stimulated by trade, allowed it to prosper
60.
Venice Biennale
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The Venice Biennale is an arts organization based in Venice, and also the original and principal exhibition it organizes. The organization changed its name to the Biennale Foundation in 2009, while the exhibition is called the Art Biennale to distinguish it from the organisation. The Art Biennale, a visual art exhibition, is so called as it is held biennially. A year later, the council decreed to adopt a by invitation system, to reserve a section of the Exhibition for foreign artists too, to works by uninvited Italian artists. The first Biennale, I Esposizione Internazionale dArte della Città di Venezia was opened on April 30,1895 by the Italian King and Queen, Umberto I, the first exhibition was seen by 224,000 visitors. In 1910 the first internationally well-known artists were displayed- a room dedicated to Gustav Klimt, a show for Renoir. A work by Picasso was removed from the Spanish salon in the central Palazzo because it was feared that its novelty might shock the public, by 1914 seven pavilions had been established, Belgium, Hungary, Germany, Great Britain, France, and Russia. During World War I, the 1916 and 1918 events were cancelled, in 1920 the post of mayor of Venice and president of the Biennale was split. The new secretary general, Vittorio Pica brought about the first presence of avant-garde art,1922 saw an exhibition of sculpture by African artists. Between the two World Wars, many important modern artists had their work exhibited there, in 1928 the Istituto Storico dArte Contemporanea opened, which was the first nucleus of archival collections of the Biennale. In 1930 its name was changed into Historical Archive of Contemporary Art, in 1933 the Biennale organised an exhibition of Italian art abroad. From 1938, Grand Prizes were awarded in the art exhibition section, during World War II, the activities of the Biennale were interrupted,1942 saw the last edition of the events. The Film Festival restarted in 1946, the Music and Theatre festivals were resumed in 1947, the Art Biennale was resumed in 1948 with a major exhibition of a recapitulatory nature. Peggy Guggenheim was invited to exhibit her famous New York collection,1949 saw the beginning of renewed attention to avant-garde movements in European—and later worldwide—movements in contemporary art. Abstract expressionism was introduced in the 1950s, and the Biennale is credited with importing Pop Art into the canon of art history by awarding the top prize to Robert Rauschenberg in 1964. From 1948 to 1972, Italian architect Carlo Scarpa did a series of interventions in the Biennales exhibition spaces. In 1954 the island San Giorgio Maggiore provided the venue for the first Japanese Noh theatre shows in Europe,1956 saw the selection of films following an artistic selection and no longer based upon the designation of the participating country. The 1957 Golden Lion went to Satyajit Rays Aparajito which introduced Indian cinema to the West,1962 included Arte Informale at the Art Exhibition with Jean Fautrier, Hans Hartung, Emilio Vedova, and Pietro Consagra
61.
Lithograph
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Lithography is a method of printing originally based on the immiscibility of oil and water. The printing is from a stone or a plate with a smooth surface. It was invented in 1796 by German author and actor Alois Senefelder as a method of publishing theatrical works. Lithography can be used to print text or artwork onto paper or other suitable material, Lithography originally used an image drawn with oil, fat, or wax onto the surface of a smooth, level lithographic limestone plate. The stone was treated with a mixture of acid and gum arabic, when the stone was subsequently moistened, these etched areas retained water, an oil-based ink could then be applied and would be repelled by the water, sticking only to the original drawing. The ink would finally be transferred to a paper sheet. This traditional technique is used in some fine art printmaking applications. In modern lithography, the image is made of a coating applied to a flexible aluminum plate. The image can be printed directly from the plate, or it can be offset, by transferring the image onto a sheet for printing. In fact, photolithography is used synonymously with offset printing, the technique as well as the term were introduced in Europe in the 1850s. Beginning in the 1960s, photolithography has played an important role in the fabrication, Lithography uses simple chemical processes to create an image. For instance, the part of an image is a water-repelling substance. Thus, when the plate is introduced to a printing ink and water mixture, the ink will adhere to the positive image. This allows a flat print plate to be used, enabling much longer, Lithography was invented by Alois Senefelder in the Kingdom of Bavaria in 1796. In the early days of lithography, a piece of limestone was used. After the oil-based image was put on the surface, a solution of gum arabic in water was applied, during printing, water adhered to the gum arabic surfaces and was repelled by the oily parts, while the oily ink used for printing did the opposite. Lithography works because of the repulsion of oil and water. The image is drawn on the surface of the print plate with a fat or oil-based medium such as a wax crayon, which may be pigmented to make the drawing visible
62.
M. C. Escher
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Maurits Cornelis Escher, or commonly M. C. Escher, was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. Apart from being used in a variety of papers, his work has appeared on the covers of many books. He was one of the inspirations of Douglas Hofstadters 1979 book Gödel, Escher. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland and he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as Mauk, he was a sickly child, although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old, in 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and he briefly studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, in the same year he traveled through Spain, visiting Madrid, Toledo, and Granada. He was impressed by the Italian countryside, and in Granada by the Moorish architecture of the fourteenth-century Alhambra, Escher returned to Italy, and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, the couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two sons, Arthur and Jan. He travelled frequently, visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra, the sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba and this turned out to be the last of his long study journeys, after 1937, his artworks were created in his studio rather than in the field. All the same, even his work already shows his interest in the nature of space, the unusual, perspective. In 1935, the climate in Italy became unacceptable to Escher
63.
Gravitation (M. C. Escher)
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Gravitation is a mixed media work by the Dutch artist M. C. It was first printed as a lithograph and then coloured by hand in watercolour. It depicts a regular polyhedron known as the small stellated dodecahedron. Each facet of the figure has a trapezoidal doorway, out of these doorways protrude the heads and legs of twelve turtles without shells, who are using the object as a common shell. The turtles are in six coloured pairs with each turtle directly opposite its counterpart
64.
Polyhedral compound
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A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called the convex hull, the compound is a facetting of the convex hull. Another convex polyhedron is formed by the central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations, a regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. There are five regular compounds of polyhedra, best known is the compound of two tetrahedra, often called the stella octangula, a name given to it by Kepler. The vertices of the two tetrahedra define a cube and the intersection of the two an octahedron, which shares the same face-planes as the compound, thus it is a stellation of the octahedron, and in fact, the only finite stellation thereof. The stella octangula can also be regarded as a dual-regular compound, the compound of five tetrahedra comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra. Each of the compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra. There are five such compounds of the regular polyhedra, the tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula. The cube-octahedron and dodecahedron-icosahedron dual compounds are the first stellations of the cuboctahedron and icosidodecahedron, the compound of the small stellated dodecahedron and great dodecahedron looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside. For this reason, the image shown above shows the small stellated dodecahedron in wireframe, in 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds made from uniform polyhedra with rotational symmetry. This list includes the five regular compounds above, the 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element, some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron. If the definition of a polyhedron is generalised they are uniform. The section for entianomorphic pairs in Skillings list does not contain the compound of two great snub dodecicosidodecahedra, as the faces would coincide. Removing the coincident faces results in the compound of twenty octahedra, in 4-dimensions, there are a large number of regular compounds of regular polytopes. There are eighteen two-parameter families of regular tessellations of the Euclidean plane
65.
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
66.
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
67.
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