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Binary Tiling
In geometry, the binary tiling (sometimes called the Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied mathematically in 1974 by . However, a closely related tiling was used earlier in a 1957 print by M. C. Escher. See especially text describing ''Regular Division of the Plane VI'', pp. 112 & 114, schematic diagram, p. 116, and reproduction of the print, p. 117. Tiles In one version of the tiling, the tiles are shapes bounded by three congruent horocyclic segments (two of which are part of the same horocycle), and two line segments. All tiles are congruent. In the Poincaré half-plane model, the horocyclic segments are modeled as horizontal line segments (parallel to the boundary of the half-plane) and the line segments are modeled as vertical line segments (perpendicular to the boundary of the half-plane), giving each tile the overall shape in the model of a square or r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tiling
In geometry, the binary tiling (sometimes called the Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied mathematically in 1974 by . However, a closely related tiling was used earlier in a 1957 print by M. C. Escher. See especially text describing ''Regular Division of the Plane VI'', pp. 112 & 114, schematic diagram, p. 116, and reproduction of the print, p. 117. Tiles In one version of the tiling, the tiles are shapes bounded by three congruent horocyclic segments (two of which are part of the same horocycle), and two line segments. All tiles are congruent. In the Poincaré half-plane model, the horocyclic segments are modeled as horizontal line segments (parallel to the boundary of the half-plane) and the line segments are modeled as vertical line segments (perpendicular to the boundary of the half-plane), giving each tile the overall shape in the model of a square or r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heesch's Problem
In geometry, the Heesch number of a shape is the maximum number of layers of copies of the same shape that can surround it. Heesch's problem is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 (the union of a square, equilateral triangle, and 30-60-90 right triangle) and proposed the more general problem. For example, a square may be surrounded by infinitely many layers of congruent squares in the square tiling, while a circle cannot be surrounded by even a single layer of congruent circles without leaving some gaps. The Heesch number of the square is infinite and the Heesch number of the circle is zero. In more complicated examples, such as the one shown in the illustration, a polygonal tile can be surrounded by several layers, but not by infinitely many; the maximum number of layers is the tile's Heesch number. Formal definitions A tessellation of the plane is a part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tilings
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] |
Hyperbolic Tree
A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level ''n'' is 2''n'', while the number of nodes for trees with more branching grows much more quickly. Drawing the tree as a node-link diagram thus requires exponential amounts of space to be displayed. One approach is to use a ''hyperbolic tree'', first introduced by Lamping et al. Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Problem
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles, that is, a shape that can tessellate space, but only in a nonperiodic way. Such a shape is called an "einstein" (not to be confused with the physicist Albert Einstein), a play on the German words ''ein Stein'', meaning ''one tile''. Depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, the problem is either open or solved. The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically. Proposed solutions In 1988, Peter Schmitt discovered a single ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (''L'', ''S'', ''R''), where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baumslag–Solitar Group
In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. They are given by the group presentation : \left \langle a, b \ : \ b a^m b^ = a^n \right \rangle. For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation. Some of the various are well-known groups. is the free abelian group on two generators, and is the fundamental group of the Klein bottle. The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non- Hopfian groups. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups. Linear representation Define :A= \begin1&1\\0&1\end, \qquad B= \begin\frac&0\\0&1\end. The matrix group generated by and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network. It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 Septe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics & Theoretical Computer Science
''Discrete Mathematics & Theoretical Computer Science'' is a peer-reviewed open access scientific journal covering discrete mathematics and theoretical computer science. It was established in 1997 by Daniel Krob (Paris Diderot University). Since 2001, the editor-in-chief is Jens Gustedt (Institut National de Recherche en Informatique et en Automatique). Abstracting and indexing The journal is abstracted and indexed in ''Mathematical Reviews'' and the Science Citation Index Expanded. According to the ''Journal Citation Reports'', the journal has a 2011 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.465. References External links * {{DEFAULTSORT:Discrete Mathematics and Theoretical Computer Science Combinatorics journals Computer science journals Pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Tiling
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice versa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Tiling
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
