1.
Science Museum, London
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The Science Museum is a major museum on Exhibition Road in South Kensington, London. It was founded in 1857 and today is one of the major tourist attractions. Like other publicly funded museums in the United Kingdom, the Science Museum does not charge visitors for admission. Temporary exhibitions, however, may incur an admission fee and it is part of the Science Museum Group, having merged with the Museum of Science and Industry in Manchester in 2012. It included a collection of machinery which became the Museum of Patents in 1858, and this collection contained many of the most famous exhibits of what is now the Science Museum. In 1883, the contents of the Patent Office Museum were transferred to the South Kensington Museum, in 1885, the Science Collections were renamed the Science Museum and in 1893 a separate director was appointed. The Art Collections were renamed the Art Museum, which became the Victoria. When Queen Victoria laid the stone for the new building for the Art Museum, she stipulated that the museum be renamed after herself. On 26 June 1909 the Science Museum, as an independent entity, the Science Museums present quarters, designed by Sir Richard Allison, were opened to the public in stages over the period 1919–28. This building was known as the East Block, construction of which began in 1913, as the name suggests it was intended to be the first building of a much larger project, which was never realized. It also contains hundreds of interactive exhibits, a recent addition is the IMAX 3D Cinema showing science and nature documentaries, most of them in 3-D, and the Wellcome Wing which focuses on digital technology. Entrance has been free since 1 December 2001, the museum houses some of the many objects collected by Henry Wellcome around a medical theme. The fourth floor exhibit is called Glimpses of Medical History, with reconstructions, the fifth floor gallery is called Science and the Art of Medicine, with exhibits of medical instruments and practices from ancient days and from many countries. The collection is strong in clinical medicine, biosciences and public health, the museum is a member of the London Museums of Health & Medicine. The Science Museum has a library, and until the 1960s was Britains National Library for Science, Medicine. It holds runs of periodicals, early books and manuscripts, and is used by scholars worldwide and it was, for a number of years, run in conjunction with the Library of Imperial College, but in 2007 the Library was divided over two sites. The Imperial College library catalogue search system now informs searchers that volumes formerly held there are Available at Science Museum Library Swindon Currently unavailable, a new Research Centre with library facilities is promised for late 2015 but is unlikely to have book stacks nearby. The Science Museums medical collections have a scope and coverage
2.
Small snub icosicosidodecahedron
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In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces,180 edges, and 60 vertices and its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß, the 40 non-sub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries and its convex hull is a nonuniform truncated icosahedron. Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the permutations of where ϕ = /2 is the golden ratio. List of uniform polyhedra Small retrosnub icosicosidodecahedron Weisstein, Eric W
3.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
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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
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Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two
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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
7.
Quasiregular polyhedron
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In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive, there are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. These forms representing a pair of a figure and its dual can be given a vertical Schläfli symbol or r to represent their containing the faces of both the regular and dual regular. A quasiregular polyhedron with this symbol will have a vertex configuration p. q. p. q, more generally, a quasiregular figure can have a vertex configuration r, representing r instances of the faces around the vertex. Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling,2. Or more generally,2, with 1/p+1/q<1/2, a regular figure with Schläfli symbol can be quasiregular, with vertex configuration q/2, if q is even. The octahedron can be considered quasiregular as a tetratetrahedron,2, similarly the square tiling 2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces,3, Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r. In this form it is known as the tetratetrahedron. The remaining convex polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram Each of these forms the core of a dual pair of regular polyhedra. The names of two of these clues to the associated dual pair, respectively the cube + octahedron. The octahedron is the core of a pair of tetrahedra. Each of these quasiregular polyhedra can be constructed by an operation on either regular parent, truncating the edges fully. This sequence continues as the tiling, vertex figure 2 - a quasiregular tiling based on the triangular tiling. But not everybody uses this terminology and these duals are transitive on their edges and faces, they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above, The rhombic dodecahedron, the rhombic triacontahedron, with two types of alternating vertices,20 with three rhombic faces, and 12 with five rhombic faces. In addition, by duality with the octahedron, the cube and their face configuration are of the form V3. n.3. n, and Coxeter-Dynkin diagram These three quasiregular duals are also characterised by having rhombic faces
8.
Density (polytope)
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In geometry, the density of a polytope represents the number of windings of a polytope, particularly a uniform or regular polytope, around its center. It can be determined by counting the minimum number of facet or face crossings of a ray from the center to infinity. The density is constant across any continuous interior region of a polytope that crosses no facets, for a non-self-intersecting polytope, the density is 1. Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point. The density of a polygon is the number of times that the polygonal boundary winds around its center. For a regular polygon, the density is q. It can be determined by counting the minimum number of edge crossings of a ray from the center to infinity. This implies a density of 7, the unmodified Eulers polyhedron formula fails for the small stellated dodecahedron and its dual great dodecahedron, for which V − E + F = −6. The regular star polyhedra exist in two pairs, with each figure having the same density as its dual, one pair has a density of 3. Hess further generalised the formula for star polyhedra with different kinds of face, the resulting value for density corresponds to the number of times the associated spherical polyhedron covers the sphere. This allowed Coxeter et al. to determine the densities of the majority of the uniform polyhedra, for hemipolyhedra, some of whose faces pass through the center, the density cannot be defined. Non-orientable polyhedra also do not have well-defined densities, there are 10 regular star polychora or 4-polytopes, which have densities between 4,6,20,66,76, and 191. They come in pairs, with the exception of the self-dual density-6. Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Coxeter, H. S. M. Longuet-Higgins, M. S. Miller, uniform polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446 Wenninger, Magnus J. An introduction to the notion of polyhedral density, Spherical models, CUP Archive, pp. 132–134, ISBN 978-0-521-22279-2 Weisstein, Eric W. Polygon density
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Spherical polyhedron
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In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the ball, thought of as a hosohedron. Some improper polyhedra, such as the hosohedra and their duals the dihedra, in the examples below, is a hosohedron and is the dual dihedron. The first known man-made polyhedra are spherical polyhedra carved in stone, many have been found in Scotland, and appear to date from the neolithic period. During the European Dark Age, the Islamic scholar Abū al-Wafā Būzjānī wrote the first serious study of spherical polyhedra, two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century, Coxeter used them to all but one of the uniform polyhedra. All the regular, semiregular polyhedra and their duals can be projected onto the sphere as tilings, given by their Schläfli symbol or vertex figure a. b. c. Spherical tilings allow cases that polyhedra do not, namely the hosohedra, regular figures as, and dihedra, regular figures as
