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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 hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brackets
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'right' bracket or, alternatively, an "opening bracket" or "closing bracket", respectively, depending on the directionality of the context. Specific forms of the mark include parentheses (also called "rounded brackets"), square brackets, curly brackets (also called 'braces'), and angle brackets (also called 'chevrons'), as well as various less common pairs of symbols. As well as signifying the overall class of punctuation, the word "bracket" is commonly used to refer to a specific form of bracket, which varies from region to region. In most English-speaking countries, an unqualified word "bracket" refers to the parenthesis (round bracket); in the United States, the square bracket. Various forms of brackets are used in mathematics, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded in 1734 by George II of Great Britain, George II, King of Great Britain and Elector of Electorate of Hanover, Hanover, and starting classes in 1737, the Georgia Augusta was conceived to promote the ideals of the Age of Enlightenment, Enlightenment. It is the oldest university in the state of Lower Saxony and the largest in student enrollment, which stands at around 31,600. Home to many List of Georg-August University of Göttingen people, noted figures, it represents one of Germany's historic and traditional institutions. According to an official exhibition held by the University of Göttingen in 2002, 44 Nobel Prize winners had been affiliated with the University of Göttingen as alumni, faculty members or researchers by that year alone. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Packing
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated '' packing density'', ''η'', of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called ''sphere packing'', which usually deals only with identical spheres. The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like. Densest packing In the two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Packing Density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. In packing problems, the objective is usually to obtain a packing of the greatest possible density. In compact spaces If are measurable subsets of a compact measure space and their interiors pairwise do not intersect, then the collection is a packing in and its packing density is :\eta = \frac. In Euclidean space If the space being packed is infinite in measure, such as Euclidean space, it is customary to define the density as the limit of densities exhibited in balls of larger and larger radii. If is the ball of radius centered at the origin, then the density of a packing is :\eta = \lim_\frac. Since this limit does not always exist, it is also useful to define the upper and lower densities as the limit superior and limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothed Octagon
The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the ''lowest'' maximum packing density of the plane of all centrally symmetric convex shapes. It was also independently discovered by Kurt Mahler in 1947. It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these. Construction The shape of the smoothed octagon can be derived from its packings, which place octagons at the points of a triangular lattice. The requirement that these packings have the same density no matter how the lattice and smoothed octagon are rotated relative to each other, with shapes that remain in contact with each neighboring shape, can be used to determine the shape of the corners. One of the figures shows three octagons that rotate while the area of the triangle formed by their centres remains constant, keeping th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called ''Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane ... [...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] |
Fundamental Region
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells. Hints at a general definition Given an action of a group ''G'' on a topological space ''X'' by homeomorphisms, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several preci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Polygon
In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius. These shapes are named after their prototypical example, the Reuleaux triangle, which in turn, is named after 19th-century German engineer Franz Reuleaux. The Reuleaux triangle can be constructed from an equilateral triangle by connecting each two vertices by a circular arc centered on the third vertex, and Reuleaux polygons can be formed by a similar construction from any regular polygon with an odd number of sides, or from certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. They have been applied in coinage shapes. Construction If P is a convex polygon with an odd number of sides, in which each vertex is equidistant to the two opposite vertices and closer to all other vertices, then replacing each side of P by an arc centered at its opposite vertex produces a Reuleaux polygon. As a special case, this construction i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Polygon
In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting. Examples All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square). A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the ... [...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'' =&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon Tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex ( or more) and sphere with three pentagons; the latter produces a tiling topologically equivalent to the dodecahedron. Monohedral convex pentagonal tilings Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eight edge-to-edge convex types, a result obtained independently by . Michaël Rao of the École normale supérieure de Lyon claimed in May 2017 to have found the proof that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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