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Petersen–Morley Theorem
In geometry, the Petersen–Morley theorem states that, if , , are three general skew lines in space, if , , are the lines of shortest distance respectively for the pairs , and , and if , and are the lines of shortest distance respectively for the pairs , and , then there is a single line meeting at right angles all of , , and . The theorem is named after Johannes Hjelmslev (who published his work on this result under his original name Johannes Trolle Petersen) and Frank Morley Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebr .... References * * * {{DEFAULTSORT:Petersen-Morley theorem Mathematical theorems Theorems in geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Lines
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. General position If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three points have been chosen, the fourth point will define a non-skew line if, and only if, it is coplanar with the first three points. However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero. If it does not, the lines defined by the points will be skew. Similarly, in three-dimensional space a very small perturbati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Hjelmslev
Johannes Trolle Hjelmslev (; 7 April 1873 – 16 February 1950) was a mathematician from Hørning, Denmark. Hjelmslev worked in geometry and history of geometry. He was the discoverer and eponym of the Hjelmslev transformation, a method for mapping an entire hyperbolic plane into a circle with a finite radius. He was the father of Louis Hjelmslev. Originally named Johannes Trolle Petersen, he changed his patronymic to the surname Hjelmslev to avoid confusion with Julius Petersen. Some of his results are known under his original name, including the Petersen–Morley theorem In geometry, the Petersen–Morley theorem states that, if , , are three general skew lines in space, if , , are the lines of shortest distance respectively for the pairs , and , and if , and are the lines of shortest distance respectively for .... Publications *Johannes Hjelmslev, ''Grundprinciper for den infinitesimale Descriptivgeometri med Anvendelse paa Læren om variable Figurer. Afhandling for d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Morley
Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebrated Morley's trisector theorem in elementary plane geometry. He led 50 Ph.D.'s to their degrees, and was said to be: :"...one of the more striking figures of the relatively small group of men who initiated that development which, within his own lifetime, brought Mathematics in America from a minor position to its present place in the sun." Life Morley was born in the town of Woodbridge in Suffolk, England. His parents were Elizabeth Muskett and Joseph Roberts Morley, Quakers who ran a china shop. After being educated at Woodbridge School, Morley went on to King's College, Cambridge (B.A., 1884). In 1887, Morley moved to Pennsylvania. He taught at Haverford College until 1900, when he became chairman of the mathematics department at J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Theorems
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
