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Biggest Little Polygon
In geometry, the biggest little polygon for a number ''n'' is the ''n''-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one ''n''-gons. One non-unique solution when ''n'' = 4 is a square, and the solution is a regular polygon when ''n'' is an odd number, but the solution is irregular otherwise. Quadrilaterals For ''n'' = 4, the area of an arbitrary quadrilateral is given by the formula ''S'' = ''pq'' sin(''θ'')/2 where ''p'' and ''q'' are the two diagonals of the quadrilateral and ''θ'' is either of the angles they form with each other. In order for the diameter to be at most 1, both ''p'' and ''q'' must themselves be at most 1. Therefore, the quadrilateral has largest area when the three factors in the area formula are individually maximized, with ''p'' = ''q'' = 1 and sin(''θ'') = 1. The condition that ''p'' = '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biggest Little Polygon
In geometry, the biggest little polygon for a number ''n'' is the ''n''-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one ''n''-gons. One non-unique solution when ''n'' = 4 is a square, and the solution is a regular polygon when ''n'' is an odd number, but the solution is irregular otherwise. Quadrilaterals For ''n'' = 4, the area of an arbitrary quadrilateral is given by the formula ''S'' = ''pq'' sin(''θ'')/2 where ''p'' and ''q'' are the two diagonals of the quadrilateral and ''θ'' is either of the angles they form with each other. In order for the diameter to be at most 1, both ''p'' and ''q'' must themselves be at most 1. Therefore, the quadrilateral has largest area when the three factors in the area formula are individually maximized, with ''p'' = ''q'' = 1 and sin(''θ'') = 1. The condition that ''p'' = '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Reinhardt (mathematician)
Karl August Reinhardt (27 January 1895 – 27 April 1941) was a German mathematician whose research concerned geometry, including polygons and tessellations. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the Reinhardt polygons. Life Reinhardt was born on January 27, 1895, in Frankfurt, the descendant of farming stock. One of his childhood friends was mathematician Wilhelm Süss. After studying at the gymnasium there, he became a student at the University of Marburg in 1913 before his studies were interrupted by World War I. During the war, he became a soldier, a high school teacher, and an assistant to mathematician David Hilbert at the University of Göttingen. Reinhardt completed his Ph.D. at Goethe University Frankfurt in 1918. His dissertation, ''Über die Zerlegung der Ebene in Polygone'', concerned tessellations of the plane, and was supervised by Ludwig Bieberbach. He began working as a secondary school teacher while working on his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Types Of Polygons
Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * Type (Unix), a command in POSIX shells that gives information about commands. * Type safety, the extent to which a programming language discourages or prevents type errors. * Type system, defines a programming language's response to data types. Mathematics * Type (model theory) * Type theory, basis for the study of type systems * Arity or type, the number of operands a function takes * Type, any proposition or set in the intuitionistic type theory * Type, of an entire function ** Exponential type Biology * Type (biology), which fixes a scientific name to a taxon * Dog type, categorization by use or function of domestic dogs Lettering * Type is a design concept for lettering used in typography which helped bring about modern textual printin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reinhardt Polygon
In geometry, a Reinhardt polygon is an equilateral polygon inscribed in a Reuleaux polygon. As in the regular polygons, each vertex of a Reinhardt polygon participates in at least one defining pair of the diameter of the polygon. Reinhardt polygons with n sides exist, often with multiple forms, whenever n is not a power of two. Among all polygons with n sides, the Reinhardt polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. They are named after Karl Reinhardt, who studied them in 1922. Definition and construction A Reuleaux polygon is a convex shape with circular-arc sides, each centered on a vertex of the shape and all having the same radius; an example is the Reuleaux triangle. These shapes are curves of constant width. Some Reuleaux polygons have side lengths that are irrational multiples of each other, but if a Reuleaux polygon has sides that can be partitioned ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hansen's Small Octagon
The largest small octagon is the octagon that has the largest area among all convex octagons with unit diameter. The diameter of a polygon is the length of the longest segment joining two of its vertices. The exact value of the area of the largest small octagon lies between 0.72686845 and 0.72686849, and is approximately 2.8% larger than the area of the regular octagon. This octagon was found in 2002 using global optimization algorithms. The optimal hexagon was found in 1975 by finding the roots of a degree-10 polynomial. See also * Biggest little polygon In geometry, the biggest little polygon for a number ''n'' is the ''n''-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one ''n''-gons. ... References External links 2010 Toulouse Global Optimization Workshopat Wolfram MathWorld Combinatorics Types of polygons {{geometry-stub ... [...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] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanfried Lenz
Hanfried Lenz (22 April 1916 in Munich1 June 2013 in Berlin) was a German mathematician, who is mainly known for his work in geometry and combinatorics. Hanfried Lenz was the eldest son of Fritz Lenz an influential German geneticist, who is associated with Eugenics and hence also with the Nazi racial policies during the Third Reich. He was also the older brother of Widukind Lenz, a geneticist. He started to study mathematics and physics at the University of Tübingen, but interrupted his studies from 1935 to 1937 to do his military service. After that he continued to study in Munich, Berlin and Leipzig. In 1939 when World War II broke out in Europe, he became a soldier in the western front and during a vacation he passed the exams for his teacher certification. He married Helene Ranke in 1943 and 1943–45 he worked on radar technology in a laboratory near Berlin. After World War II Hanfried Lenz was classified as a "follower" by the denazification process. He started to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Graham
Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences. After graduate study at the University of California, Berkeley, Graham worked for many years at Bell Labs and later at the University of California, San Diego. He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness, and many topics in mathematics are named after him. He published six books and about 400 papers, and had nearly 200 co-authors, including many collaborative works with his wife Fan Chung and with Paul Erdős. Graham has been featured in ''Ripley's Believe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthodiagonal Quadrilateral
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other. Special cases A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals. A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram). A square is a limiting case of both a kite and a rhombus. Orthodiagonal equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the ''n'' = 4 case of the biggest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
