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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] |
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 geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kőnig's Lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theorem also has important roles in constructive mathematics and proof theory. Statement of the lemma Let G be a connected, locally finite, infinite graph. This means that every two vertices can be connected by a finite path, the graph has infinitely many vertices, and each vertex is adjacent to only finitely many other vertices. Then G contains a ray: a simple path (a path with no repeated vertices) that starts at one vertex and continues from it through infinitely many vertices. A useful special case of the lemma is that every infinite tree contains either a vertex of infinite degree or an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon With Heesch Number 6
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its ''edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bojan Bašić
Bojan (Serbian Cyrillic and Macedonian: Бојан; Ukrainian, Russian and Bulgarian Cyrillic: Боян, transcribed ''Boyan'') is a Slavic given name, derived from the Slavic noun ''boj'' "battle." The ending ''-an'' is a suffix frequently found in anthroponyms of Slavic origin. The feminine variant is Bojana. The name is recorded in historical sources among Serbs, Bulgarians, Czechs, Poles, Croats, Slovenians, Macedonians, Ukrainians and Russians. In Slovenia, it is the 18th most popular name for males, as of 2010. (in Slovenian). Statistical Office of the Republic of Slovenia. The name Bojan may refer to: * |
Casey Mann
Casey Mann is an American mathematician, specializing in discrete and computational geometry, in particular tessellation and knot theory. He is Professor of Mathematics at University of Washington Bothell, and received the PhD at the University of Arkansas in 2001. He is known for his 2015 discovery, with Jennifer McLoud-Mann and undergraduate student David Von Derau, of the 15th and last class of convex pentagons to tile the plane. Mann is also known for his work on Heesch's problem, to which he contributed a polygon with ''Heesch number'' 5. This problem is closely related to the einstein problem, of whether there exists a shape that can tessellate space, but only in a non-periodic way. Education and career Mann received his B.S. in Mathematics at East Central University in Ada, Oklahoma, and completed his Ph.D. in 2001 from the University of Arkansas. His dissertation in discrete geometry, supervised by Chaim Goodman-Strauss, was ''Heesch's Problem and Other Tiling Proble ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anne Fontaine (Mathematician)
Anne Fontaine (born Anne-Fontaine Sibertin-Blanc; 15 July 1959) is a Luxembourger film director, screenwriter, and former actress. She lives and works in France. Life and career Born Anne-Fontaine Sibertin-Blanc in Luxembourg, sister of actor Jean-Chrétien Sibertin-Blanc, she went as a young child to live in Lisbon, where her father, Antoine Sibertin-Blanc, is a music professor and cathedral organist. In adolescence she moved to Paris and trained in dance with Joseph Russillo while continuing her academic education, including philosophy. Her husband is Philippe Carcassonne, the film producer, and they have an adopted son, Tienne, who was born in Cambodia. While still dancing, she was picked by Robert Hossein to play Esmeralda in a 1980 theatrical production of ''The Hunchback of Notre Dame'' and around this time started to use the name Anne Fontaine. She continued with acting and became known for her roles in comedies like ''Si ma gueule vous plaît...'' (1981) and ''P.R.O.F.S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in '' Fairy Chess Review'' between the years 1937 to 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Ammann
Robert Ammann (October 1, 1946 – May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann attended Brandeis University, but generally did not go to classes, and left after three years. He worked as a programmer for Honeywell. After ten years, his position was eliminated as part of a routine cutback, and Ammann ended up working as a mail sorter for a post office. In 1975, Ammann read an announcement by Martin Gardner of new work by Roger Penrose. Penrose had discovered two simple sets of aperiodic tiles, each consisting of just two quadrilaterals. Since Penrose was taking out a patent, he wasn't ready to publish them, and Gardner's description was rather vague. Ammann wrote a letter to Gardner, describing his own work, which duplicated one of Penrose's sets, plus a foursome of "golden rhombohedra" that formed aperiodic tilings in space. More letters followed, and Ammann be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich Heesch
Heinrich Heesch (June 25, 1906 – July 26, 1995) was a German mathematician. He was born in Kiel and died in Hanover. In Göttingen he worked on Group theory. In 1933 Heesch witnessed the National Socialist purges of university staff. Not willing to become a member of the National Socialist organization of university teachers as required, he resigned from his university position in 1935 and worked privately at his parents' home in Kiel until 1948. During this time he did research on tilings. In 1955 Heesch began teaching at Leibniz University Hannover and worked on graph theory. In this period Heesch did pioneering work in developing methods for a computer-aided proof of the then unproved four color theorem. In particular, he was the first to investigate the notion of "discharging", which turned out to be a fundamental ingredient of the eventual computer-aided proof by Kenneth Appel and Wolfgang Haken. Between 1967 and 1971, Heesch made several visits to the United ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
