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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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Space
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and '' symbolic dynamical systems'' are often considered synonyms. The most widely studied shift spaces are the subshifts of finite type. Notation Let ''A'' be a finite set of states. An ''infinite'' (respectively ''bi-infinite'') ''word'' over ''A'' is a sequence \mathbf x=(x_n)_, where M=\mathbb N (respectively M=\mathbb Z) and x_n is in ''A'' for any n \in M. The shift operator \sigma acts on an infinite or bi-infinite word by shifting all symbols to the left, i.e., :\sigma(\mathbf x)_n=x_ for all ''n''. In the following we choose M=\mathbb N and thus speak of infinite words, but all definitions are naturally generalizable to the bi-infinite case. Definition A set of infinite words over ''A'' is a ''shift space'' (or ''subshift'') if it is closed with respect to the natural product topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Federation Square
Federation Square (colloquially Fed Square) is a venue for arts, culture and public events on the edge of the Melbourne central business district. It covers an area of at the intersection of Flinders and Swanston Streets built above busy railway lines and across the road from Flinders Street station. It incorporates major cultural institutions such as the Ian Potter Centre, Australian Centre for the Moving Image (ACMI) and the Koorie Heritage Trust as well as cafes and bars in a series of buildings centred around a large paved square, and a glass walled atrium. History Background Melbourne's central city grid was originally designed without a central public square, long seen as a missing element. From the 1920s, there had been proposals to roof the railway yards on the south-east corner of Flinders and Swanston Streets for a public square, with more detailed proposals prepared in the 1950s and 1960s. In the 1960s, the Melbourne City Council decided that the best place for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinwheel Fractal
Pinwheel may refer to: * Pinwheel (toy), a spinning children's toy * Pinwheel (cryptography), a device for producing a short pseudo-random sequence of bits * Pinwheel (shogi), an opening in the game shogi or Japanese chess * Pinwheel (TV channel), a channel which would later turn into Nickelodeon * ''Pinwheel'' (TV series), a children's show on Nickelodeon that ran from 1977 to 1984 * Pinwheel calculator (part of), a type of early mechanical arithmetic machine * ''Tabernaemontana divaricata'', also known as pinwheel flower * Pinwheel tilings, aperiodic tilings of the plane whose tiles appear in infinitely many orientations * Catherine wheel (firework), a form of pyrotechnic display device also known as a pinwheel * ''Coenocharopa elegans'', also known as the elegant pinwheel snail, a land snail found in Queensland, Australia * "Pinwheels", a poem by Patti Smith from her 1978 book ''Babel'' * Pinwheel USY, part of United Synagogue Youth covering the Pacific Northwest * Wartenberg wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete And Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaquaversal Tiling
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern that relies essentially on their previous construct, the pinwheel tiling. The rotations relating these tiles belong to the group G(6,4) generated by two rotations of order 6 and 4 whose axes are perpendicular to each other. These rotations are dense in SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr .... References *. *. External links * pictureof a quaquaversal tiling Charles Radinpage at the University of Texas Discrete geometry Tessellation {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Dynamics
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another. History The idea goes back to Jacques Hadamard's 1898 paper on the geodesics on surfaces of negative curvature. It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by Emil Artin in 1924 (for the system now called Artin billiard), Pekka Myrberg, Paul Koebe, Jakob Nielsen, G. A. Hedlund. The first formal treatment was developed by Morse and Hedlund in their 1938 paper. George Birkh ... [...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 ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinwheel 3
Pinwheel may refer to: * Pinwheel (toy), a spinning children's toy * Pinwheel (cryptography), a device for producing a short pseudo-random sequence of bits * Pinwheel (shogi), an opening in the game shogi or Japanese chess * Pinwheel (TV channel), a channel which would later turn into Nickelodeon * ''Pinwheel'' (TV series), a children's show on Nickelodeon that ran from 1977 to 1984 * Pinwheel calculator (part of), a type of early mechanical arithmetic machine * ''Tabernaemontana divaricata'', also known as pinwheel flower * Pinwheel tilings, aperiodic tilings of the plane whose tiles appear in infinitely many orientations * Catherine wheel (firework), a form of pyrotechnic display device also known as a pinwheel * ''Coenocharopa elegans'', also known as the elegant pinwheel snail, a land snail found in Queensland, Australia * "Pinwheels", a poem by Patti Smith from her 1978 book ''Babel'' * Pinwheel USY, part of United Synagogue Youth covering the Pacific Northwest * Wartenberg wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
