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Mathematical Models (Fischer)
''Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary'' is a book on the physical models of concepts in mathematics that were constructed in the 19th century and early 20th century and kept as instructional aids at universities. It credits Gerd Fischer as editor, but its photographs of models are also by Fischer. It was originally published by Vieweg+Teubner Verlag for their bicentennial in 1986, both in German (titled ''Mathematische Modelle. Aus den Sammlungen von Universitäten und Museen. Mit 132 Fotografien. Bildband und Kommentarband'') and (separately) in English translation, in each case as a two-volume set with one volume of photographs and a second volume of mathematical commentary. Springer Spektrum reprinted it in a second edition in 2017, as a single dual-language volume. Topics The work consists of 132 full-page photographs of mathematical models, divided into seven categories, and seven chapters of mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dupin Cyclide
In mathematics, a Dupin cyclide or cyclide of Dupin is any Inversive geometry, geometric inversion of a standard torus, Cylinder (geometry), cylinder or cone, double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after) Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one-parameter family of spheres) in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry. Dupin cyclides are often simply known as ''cyclides'', but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions. Dupin cyclides were investigated not only by Dupin, but also by Arthur Cayley, A. Cayley, James Clerk Maxwell, J.C. Maxwell and Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A Function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a Plane curve, curve. The graphical representation of the graph of a Function (mathematics), function is also known as a ''Plot (graphics), plot''. In the case of Bivariate function, functions of two variables – that is, functions whose Domain of a function, domain consists of pairs (x, y) –, the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a Surface (mathematics), surface, which can be visualized as a ''surface plot (graphics), surface plot''. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family. Hopf attended Karl Mittelhaus higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the Silesian Friedrich Wilhelm University where he attended lectures by Ernst Steinitz, Adolf Kneser, Max Dehn, Erhard Schmidt, and Rudolf Sturm. When World War I broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918. After the war Hopf continued his mathematical education in Heidelberg (winter 1919/2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Apéry
Roger Apéry (; 14 November 1916, Rouen – 18 December 1994, Caen) was a Greek-French mathematician most remembered for Apéry's theorem, which states that is an irrational number. Here, denotes the Riemann zeta function. Biography Apéry was born in Rouen in 1916 to a French mother and Greek father. His childhood was spent in Lille until 1926, when the family moved to Paris, where he studied at the Lycée Ledru-Rollin and the Lycée Louis-le-Grand. He was admitted at the École normale supérieure in 1935. His studies were interrupted at the start of World War II; he was mobilized in September 1939, taken prisoner of war in June 1940, repatriated with pleurisy in June 1941, and hospitalized until August 1941. He wrote his doctoral thesis in algebraic geometry under the direction of Paul Dubreil and René Garnier in 1947. In 1947 Apéry was appointed Maître de conférences (lecturer) at the University of Rennes. In 1949 he was appointed Professor at the University of C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boy's Surface
In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space. It was discovered in 1901 by the German mathematician Werner Boy, who had been tasked by his doctoral thesis advisor David Hilbert to prove that the projective plane ''could not'' be immersed in three-dimensional space. Boy's surface was first parametrized explicitly by Bernard Morin in 1978. Another parametrization was discovered by Rob Kusner and Robert Bryant.. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections (that is, it has no pinch-points). Parametrization Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant, is the following: given a complex number ''w'' whose magnitude is less than or equal to one ( \, w \, \le 1), let :\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Star Polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 14 quasiregular ones, and 39 semiregular ones. There are also two infinite sets of ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have polygon density greater than 1) correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Tetrahedron
The Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. Thus the center of each ball is on the surfaces of the other three balls. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges. This shape is defined and named by analogy to the Reuleaux triangle, a two-dimensional curve of constant width; both shapes are named after Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another. One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width, but it is not true: the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Triangle
A Reuleaux triangle is a circular triangle, curved triangle with curve of constant width, constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circle, circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?" They are named after Franz Reuleaux,. a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the designers of Gothic architecture, Gothic church windows, by Leonardo da Vinci, who used it for a O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Constant Width
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an object is more than "a mere geometric solid", but is "filled with, spread over by, or suffused with perceivable qualities such as color and warmth". The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several inequivalent such formalizations that are all called ''surface'', sometimes with a qualifier such as algebraic surface, smooth surface or fractal surface. The concept of surface and its mathematical abstractions are both widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
