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Elliptic Geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as ''single elliptic geometry'' whereas spherical geometry is sometimes referred to as ''double elliptic geometry''. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. Definitions Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books '' Astronomia nova'', '' Harmonice Mundi'', and '' Epitome Astronomiae Copernicanae'', influencing among others Isaac Newton, providing one of the foundations for his theory of universal gravitation. The variety and impact of his work made Kepler one of the founders and fathers of modern astronomy, the scientific method, natural and modern science. He has been described as the "father of science fiction" for his novel '' Somnium''. 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 eventually the imperial mathematician to Emperor Rudolf II and his two succe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly Hamiltonian mechanics, his reformulation of Lagrangian mechanics. His career included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra. Hamilton was Andrews Professor of Astronomy at Trinity College Dublin. He was also the third director of Dunsink Observatory from 1827 to 1865. The Hamilton Institute at Maynooth University is named after him. Early life Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street, Dublin, Dominick Street, later renumbered to 36. Ham ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equipollence (geometry)
In Euclidean geometry, equipollence is a homogeneous relation between directed line segments. Two segments are said to be ''equipollent'' when they have the same length and direction. Two equipollent segments are parallel but not necessarily colinear nor overlapping. For example, a segment ''AB'', from point ''A'' to point ''B'', has the opposite direction to segment ''BA''; thus ''AB'' and ''BA'' are equipollent. Parallelogram property A property of Euclidean spaces is the parallelogram property of vectors: If two segments are equipollent, then they form two sides of a (possibly degenerate) parallelogram: History The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently, the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides , and the hypotenuse , sometimes called the Pythagorean equation: :a^2 + b^2 = c^2 . The theorem is named for the Ancient Greece, Greek philosopher Pythagoras, born around 570 BC. The theorem has been Mathematical proof, proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both Geometry, geometric proofs and Algebra, algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smoothly from point to point). This gives, in particular, local notions of angle, arc length, length of curves, surface area and volume. From those, some other global quantities can be derived by integral, integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in Three-dimensional space, R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Danne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations between geometry and group theory. His 1872 Erlangen program classified geometries by their basic symmetry groups and was an influential synthesis of much of the mathematics of the time. During his tenure at the University of Göttingen, Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His Felix Klein Protocols, seminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction and promoted mathematics education reform at all grade levels in Germany and abroad. He became the first president of the International Commission on Mathematical Instruction in 1908 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society Of London
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the second journal in the world exclusively devoted to science, after the ''Journal des sçavans'', and therefore also the world's longest-running scientific journal. It became an official society publication in 1752. The use of the word ''philosophical'' in the title refers to natural philosophy, which was the equivalent of what would now be generally called ''science''. Current publication In 1887 the journal expanded and divided into two separate publications, one serving the physical sciences (''Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'') and the other focusing on the life sciences (''Philosophical Transactions of the Royal Society B: Biological Sciences''). Both journals now publish themed i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics. Early life Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of George Cayley, the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line At Infinity
In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. Geometric formulation In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
