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Danzer Set
In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Density One way to define the problem more formally is to consider the growth rate of a set S in Euclidean space, defined as the function that maps a real number r to the number of points of S that are within distance r of the origin. Danzer's question is whether it is possible for a Danzer set to have growth expressed in big O notation. If so, this would equal the growth rate of well-spaced point sets like the integer lattice (which is not a Danzer set). It is possible to construct a Danzer set of growth rate that is within a polylogarithmic factor For instance, overlaying rectangular grids whose cells have constant volume but differing aspect ratios can achieve a growth rate Constructions for Danzer sets are known with a somewhat slower growth rate, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Danzer Set
In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Density One way to define the problem more formally is to consider the growth rate of a set S in Euclidean space, defined as the function that maps a real number r to the number of points of S that are within distance r of the origin. Danzer's question is whether it is possible for a Danzer set to have growth expressed in big O notation. If so, this would equal the growth rate of well-spaced point sets like the integer lattice (which is not a Danzer set). It is possible to construct a Danzer set of growth rate that is within a polylogarithmic factor For instance, overlaying rectangular grids whose cells have constant volume but differing aspect ratios can achieve a growth rate Constructions for Danzer sets are known with a somewhat slower growth rate, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thrackle
A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, the crossing must be ''transverse''.. A preliminary version of these results was reviewed in . Linear thrackles A linear thrackle is a thrackle drawn in such a way that its edges are straight line segments. As Paul Erdős observed, every linear thrackle has at most as many edges as vertices. If a vertex ''v'' is connected to three or more edges ''vw'', ''vx'', and ''vy'', at least one of those edges (say ''vw'') lies on a line that separates two other edges. Then, ''w'' must have degree one, because no line segment ending at ''w'', other than ''vw'', can touch both ''vx'' and ''vy''. Removing ''w'' and ''vw'' produces a smaller thrackle, without changing the difference between the numbers of edges and vertices. Aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punched cards ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski's Theorem
In mathematics, Minkowski's theorem is the statement that every convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point (meaning a point in \Z^n that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice L and to any symmetric convex set with volume greater than 2^n\,d(L), where d(L) denotes the covolume of the lattice (the absolute value of the determinant of any of its bases). Formulation Suppose that is a lattice of determinant in the - dimensional real vector space and is a convex subset of that is symmetric with respect to the origin, meaning that if is in then is also in . Minkowski's theorem states that if the volume of is strictly greater than , then must contain at least one lattice point other than the origin. (Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heilbronn Triangle Problem
In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed in a given area, the smallest triangle area will be at most inversely proportional to the square of the number of points. His conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown. Definition The Heilbronn triangle problem concerns the placement of n points within a shape in the plane, such as the unit square or the unit disk, for a given Each triple of points form the three vertices of a triangle, and among these triangles, the problem concerns the smallest triangle, as measured by area. Different placements of points will have different smallest triangles, and the problem asks: how should n points be placed to maximize the area of the smallest More formally, the shape may be assumed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penrose Tiling
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s. There are several different variations of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together in a way that avoids periodic tiling. This may be done in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinwheel Tiling
In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations. Conway's tessellation 250px, Conway's triangle decomposition into smaller similar triangles. Let T be the right triangle with side length 1, 2 and \sqrt. Conway noticed that T can be divided in five isometric copies of its image by the dilation of factor 1/\sqrt. 250px, The increasing sequence of triangles which defines Conway's tiling of the plane. By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of T. The union of all these triangles yields a tiling of the whole plane by isometric copies of T. In this tiling, isometric copies of T appear in infinitely many orientations (this is due to the angles \arc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tiling
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tiling
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution Tiling
In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid. Introduction A tile substitution is described by a set of prototiles (tile shapes) T_1,T_2,\dots, T_m, an expanding map Q and a dissection rule showing how to dissect the expanded prototiles Q T_i to form copies of some prototiles T_j. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked til ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylver Coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of '' Winning Ways for Your Mathematical Plays''. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named after James Joseph Sylvester, who proved that if ''a'' and ''b'' are relatively prime positive integers, then (''a'' − 1)(''b'' − 1) − 1 is the largest number that is not a sum of nonnegative multiples of ''a'' and ''b''. Thus, if ''a'' and ''b'' are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the greatest common divisor of the moves played so far is ''g'', then only finitely many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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