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Mathematical Models (Cundy And Rollett)
''Mathematical Models'' is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981. The vertex configuration of a uniform polyhedron, a generalization of the Schläfli symbol that describes the pattern of polygons surrounding each vertex, was devised in this book as a way to name the Archimedean solids, and has sometimes been called the ''Cundy–Rollett symbol'' as a nod to this origin. Topics The first edition of the book had five chapters, including its introduction which discusses model-making in general and the different media and tools with which one can construct models. The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal". The second chapter concerns plane geometry, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martyn Cundy
Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to reform O level and A level teaching. Through this he had a big effect on maths teaching in Britain and especially in Africa. Education and career Cundy attended Monkton Combe School and then read mathematics at Trinity College, Cambridge, where he earned a PhD in quantum theory in 1938. In 1937, Cundy was awarded the Cambridge University Rayleigh Prize for Mathematical Physics (now known as the Rayleigh-Knight Prize) for an essay entitled "Motion in a Tetrahedral Field". Others awarded the Rayleigh Prize include Royal Society Fellows Alan Turing and Fred Hoyle; instead of acquiring a University position, Cundy initially chose work at the secondary school level. He taught at the Sherborne School from 1938 to 1966 and became prominently involved in the refo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron Compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices. The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. The prismatic compounds of prisms ( UC20 and UC21) exist only when , and when and are coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv .... The uniform prismatic compounds of antiprisms ( UC22, UC23, UC24 and UC25) exist only when , and when and are coprime. Furthermore, when , the antiprisms dege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linkage (mechanical)
A mechanical linkage is an assembly of systems connected so as to manage forces and Motion, movement. The movement of a body, or link, is studied using geometry so the link is considered to be Rigid body, rigid. The connections between links are modeled as providing ideal movement, pure rotation or Sliding (motion), sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain. Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph. The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean geometry, Euclidean displacements. The number of parameters in the subgroup is called the degree of freedom (mecha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonograph
A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are Lissajous curves or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, although Hugh Blackburn, a professor of mathematics at the University of Glasgow, is commonly believed to be the official inventor. A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a pen relative to a drawing surface. One pendulum moves the pen back and forth along one axis, and the other pendulum moves the drawing surface back and forth along a perpendicular axis. By varying the frequency and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create ellipses, spirals, figure eights and other Lissajous figures. More complex harmono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Figure
A string figure is a design formed by manipulating twine, string on, around, and using one's fingers or sometimes between the fingers of multiple people. String figures may also involve the use of the mouth, wrist, and feet. They may consist of singular images or be created and altered as a game, known as a string game, or as part of a narrative, story involving various figures made in sequence (string story). String figures have also been used for divination, such as to predict the sex of an unborn child. A popular string game is cat's cradle, but many string figures are known in many places under different names, and string figures are well distributed throughout the world.Elffers, Joost and Schuyt, Michael (1978/1979). ''Cat's Cradles and Other String Figures'', p.197. . History According to Camilla Gryski, a Canadian librarian and author of numerous string figure books, "We don't know when people first started playing with string, or which primitive people invented this an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to Non-Euclidean geometry, non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the ''packing density'' of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Conjecture
In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less th ... for the number of colors that are necessary for graph coloring on a surface (topology), surface of a given genus (mathematics), genus. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required number of colors is 4, 7, 8, 9, 10, 11, 12, 12, .... , the ''chromatic number'' or ''Heawood number''. The conjecture was formulated in 1890 by Percy John Heawood, P.J. Heawood and mathematical proof, proven in 1968 by Gerhard Ringel and John William Theodore Youngs, J.W.T. Youngs. One case, the orientability, non-orientable Klein bottle, proved an exception to the general formula. An entirely different approach was needed for the much older problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Bottle
In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuous function, continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with a Boundary (topology), boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary. The Klein bottle was first described in 1882 by the mathematician Felix Klein. Construction The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Ancient Rome, Roman mosaics from the third century Common Era, CE. The Möbius strip is a orientability, non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip. As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a Knot (mathematics), knotted centerline. Any two embeddings with the same knot for the centerline and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a Lemon (geometry), spindle torus (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadric
In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hypersurface (of dimension ''D'') embedded in a higher dimensional space (of dimension ) is defined as the zero set of an irreducible polynomial of degree two in variables; for example, ''D''1 is the case of conic sections ( plane curves). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see , below. Formulation In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
