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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] |
Hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler Conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Close-packing Of Spheres
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is :\frac \approx 0.74048. The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only for 1, 2, 3, 8, and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one anoth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rye Castle, Rye, East Sussex, England-6April2011 (1) (cropped)
Rye (''Secale cereale'') is a grass grown extensively as a grain, a cover crop and a forage crop. It is a member of the wheat tribe (Triticeae) and is closely related to both wheat (''Triticum'') and barley (genus ''Hordeum''). Rye grain is used for flour, bread, beer, crispbread, some whiskeys, some vodkas, and animal fodder. It can also be eaten whole, either as boiled rye berries or by being rolled, similar to rolled oats. Rye is a cereal grain and should not be confused with ryegrass, which is used for lawns, pasture, and as hay for livestock. Distribution and habitat Rye is one of a number of species that grow wild in the Levant, central and eastern Turkey and in adjacent areas. Evidence uncovered at the Epipalaeolithic site of Tell Abu Hureyra in the Euphrates valley of northern Syria suggests that rye was among the first cereal crops to be systematically cultivated, around 13,000 years ago. However, that claim remains controversial; critics point to incons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order And Chaos
Order and Chaos is a variant of the game tic-tac-toe on a 6×6 . It was invented by Stephen Sniderman and introduced by him in ''Games'' magazine in 1981. The player ''Order'' strives to create a five-in-a-row of either Xs or Os. The opponent ''Chaos'' endeavors to prevent this. Game rules Unlike typical board games, both players control both sets of pieces (Xs and Os). The game starts with the . Order plays first, then turns alternate. On each turn, a player places either an X or an O on any open square. Once played, pieces cannot be moved, thus Order and Chaos can be played using pencil and paper. Order aims to get five like pieces in a row either vertically, horizontally, or diagonally. Chaos aims to fill the board without completion of a line of five like pieces. Rules addition The original rules in ''Games'' magazine implied that six-in-a-row also wins. That version of the game was claimed weakly solved as a forced win for Order. The inventor has subsequently suggested a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Granular Material
A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compose granular material are large enough such that they are not subject to thermal motion fluctuations. Thus, the lower size limit for grains in granular material is about 1 μm. On the upper size limit, the physics of granular materials may be applied to ice floes where the individual grains are icebergs and to asteroid belts of the Solar System with individual grains being asteroids. Some examples of granular materials are snow, nuts, coal, sand, rice, coffee, corn flakes, fertilizer, and bearing balls. Research into granular materials is thus directly applicable and goes back at least to Charles-Augustin de Coulomb, whose law of friction was originally stated for granular materials. Granular materials are commercially important in applicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jamming (physics)
Jamming is the physical process by which the viscosity of some mesoscopic materials, such as granular materials, glasses, foams, polymers, emulsions, and other complex fluids, increases with increasing particle density. The jamming transition has been proposed as a new type of phase transition, with similarities to a glass transition but very different from the formation of crystalline solids. While a glass transition occurs when the liquid state is cooled, the jamming transition happens when the density, or the packing fraction of the particles, is increased. This crowding of the constituent particles prevents them from flowing under an applied stress and from exploring phase space, thus making the aggregate material behave as a solid. The system may be able to unjam if volume fraction is decreased, or external stresses are applied such that they exceed the yield stress. This transition is interesting because it is nonlinear with respect to volume fraction. The jamming phas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Proof Checking
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Exhaustion
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: # A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. # A proof of each of the cases. The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Fejes Tóth
László Fejes Tóth ( hu, Fejes Tóth László, 12 March 1915 – 17 March 2005) was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer. He was a member of the Hungarian Academy of Sciences (from 1962) and a director of the Alfréd Rényi Institute of Mathematics (1970-1983). He received both the Kossuth Prize (1957) and State Award (1973). Together with H.S.M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry. Early life and career As described in a 1999 interview witIstván Hargittai Fejes Tóth's father was a railway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Callister Hales
Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture. Biography He received his Ph.D. from Princeton University in 1986, his dissertation was titled ''The Subregular Germ of Orbital Integrals''. Hales taught at Harvard University and the University of Chicago, and from 1993 and 2002 he worked at the University of Michigan. In 1998, Hales submitted his paper on the computer-ai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes 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''. These works also provided one of the foundations for Newton's theory of universal gravitation. 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 successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting (or Keplerian) telescope, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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