1.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry, Hyperbolic geometry can be extended to three and more dimensions, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry, For any line R and any point P which does not lie on R. In the plane containing line R and point P there are at least two lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
2.
Klein quartic
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As such, the Klein quartic is the Hurwitz surface of lowest possible genus, see Hurwitzs automorphisms theorem. Its automorphism group is isomorphic to PSL, the second-smallest non-abelian simple group, the quartic was first described in. Originally, the Klein quartic referred specifically to the subset of the projective plane P2 defined by an algebraic equation. This has a specific Riemannian metric, under which its Gaussian curvature is not constant and this gives the Klein quartic a Riemannian metric of constant curvature −1 that it inherits from H2. This group is known as PSL, and also as the isomorphic group PSL. By covering space theory, the group G mentioned above is isomorphic to the group of the compact surface of genus 3. It is important to two different forms of the quartic. The closed quartic is what is meant in geometry, topologically it has genus 3 and is a compact space. The open or punctured quartic is of interest in theory, topologically it is a genus 3 surface with 24 punctures. The open quartic may be obtained from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The Klein quartic can be viewed as an algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates on P2. The locus of this equation in P2 is the original Riemannian surface that Klein described, note the identity 3 =72, exhibiting 2 - η as a prime factor of 7 in the ring of integers. The group Γ is a subgroup of the triangle group. Namely, Γ is a subgroup of the group of elements of unit norm in the algebra generated as an associative algebra by the generators i, j. One chooses a suitable Hurwitz quaternion order Q H u r in the quaternion algebra, Γ is then the group of norm 1 elements in 1 + I Q H u r. The least absolute value of a trace of an element in Γ is η2 +3 η +2, corresponding the value 3.936 for the systole of the Klein quartic. The Klein quartic admits tilings connected with the group. This tiling is a quotient of the order-3 bisected heptagonal tiling of the hyperbolic plane and this tiling is uniform but not regular, and often regular tilings are used instead
3.
Truncated order-7 triangular tiling
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In geometry, the Order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a similar to a conventional soccer ball with heptagons in place of pentagons. It has Schläfli symbol of t and this tiling is called a hyperbolic soccerball for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a surface can be constructed in 3-space. The dual tiling is called a heptakis heptagonal tiling, named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point. This hyperbolic tiling is related as a part of sequence of uniform truncated polyhedra with vertex configurations. From a Wythoff construction there are eight uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk
4.
Systolic geometry
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See also a slower-paced Introduction to systolic geometry. The systole of a metric space X is a metric invariant of X. In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the group of X. When X is a graph, the invariant is usually referred to as the girth, possibly inspired by Tuttes article, Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student Pao Ming Pu. The actual term itself was not coined until a quarter century later. Referring to these inequalities, Thom reportedly exclaimed, Mais cest fondamental. Subsequently, Berger popularized the subject in a series of articles and books, a bibliography at the Website for systolic geometry and topology currently contains over 160 articles. Systolic geometry is a developing field, featuring a number of recent publications in leading journals. Recently, the link with the Lusternik–Schnirelmann category has emerged, the existence of such a link can be thought of as a theorem in systolic topology. Every convex centrally symmetric polyhedron P in R3 admits a pair of opposite points, an alternative formulation is as follows. Any centrally symmetric body of surface area A can be squeezed through a noose of length π A. This property is equivalent to a case of Pus inequality. To give an idea of the flavor of the field. The main thrust of Thoms remark to Berger quoted above appears to be the following, whenever one encounters an inequality relating geometric invariants, such a phenomenon in itself is interesting, all the more so when the inequality is sharp. The classical isoperimetric inequality is a good example, in systolic questions about surfaces, integral-geometric identities play a particularly important role. Roughly speaking, there is an integral identity relating area on the one hand, by the Cauchy–Schwarz inequality, energy is an upper bound for length squared, hence one obtains an inequality between area and the square of the systole. This inequality can be thought of as analogous to Bonnesens inequality with isoperimetric defect, a number of new inequalities of this type have recently been discovered, including universal volume lower bounds. More details appear at systoles of surfaces, here the homotopy systole sysπ1 is by definition the least length of a noncontractible loop in M
5.
Modular group
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In mathematics, the modular group is the projective special linear group PSL of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified, the group operation is function composition. This group of transformations is isomorphic to the special linear group PSL. In other words, PSL consists of all matrices where a, b, c, and d are integers, ad − bc =1, the group operation is the usual multiplication of matrices. Some authors define the group to be PSL, and still others define the modular group to be the larger group SL. Some mathematical relations require the consideration of the group GL of matrices with determinant plus or minus one, similarly, PGL is the quotient group GL/. A2 ×2 matrix with unit determinant is a matrix, and thus SL = Sp. The unit determinant of implies that the fractions a/b, a/c, c/d and b/d are all irreducible, more generally, if p/q is an irreducible fraction, then a p + b q c p + d q is also irreducible. Elements of the group provide a symmetry on the two-dimensional lattice. Let ω1 and ω2 be two numbers whose ratio is not real. Then the set of points Λ = is a lattice of parallelograms on the plane, a different pair of vectors α1 and α2 will generate exactly the same lattice if and only if = for some matrix in GL. It is for this reason that doubly periodic functions, such as elliptic functions, the action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point corresponding to the fraction p/q. An irreducible fraction is one that is visible from the origin, the action of the group on a fraction never takes a visible to a hidden one. If p n −1 / q n −1 and p n / q n are two successive convergents of a fraction, then the matrix belongs to GL. In particular, if bc − ad =1 for positive integers a, b, c and d with a < b and c < d then a/b, important special cases of continued fraction convergents include the Fibonacci numbers and solutions to Pells equation. In both cases, the numbers can be arranged to form a subset of the modular group. Geometrically, S represents inversion in the unit followed by reflection with respect to the imaginary axis. The generators S and T obey the relations S2 =1 and 3 =1
6.
Noam Elkies
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Noam David Elkies is an American mathematician and chess master. Along with A. O. L. Atkin, he extended Schoofs algorithm to create the Schoof–Elkies–Atkin algorithm, in 1993, when he was 26 years old, he became the youngest full professor in the history of Harvard University. He was a Putnam Fellow two more times during his undergraduate years, in 1987, he proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Eulers sum of powers conjecture for fourth powers and his work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, he was made a full, tenured professor at the age of 26 and this made him the youngest full professor in the history of Harvard. Elkies, along with A. O. L. Atkin, in 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich. In 2004 he received a Lester R. Ford Award, Elkies also studies the connections between music and mathematics. He sits on the Advisory Board of the Journal of Mathematics and he has discovered many new patterns in Conways Game of Life and has studied the mathematics of still life patterns in that cellular automaton rule. Elkies is a fellow at Harvards Lowell House, Elkies is a composer and solver of chess problems. He holds the title of National Master from the United States Chess Federation, but he no longer plays competitively
7.
Journal of Differential Geometry
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The Journal of Differential Geometry is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes a supplement in book form called Surveys in Differential Geometry. It covers differential geometry and related such as differential equations, mathematical physics, algebraic geometry. The editor-in-chief is Shing-Tung Yau of Harvard University, the journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journals editor-in-chief, and later co-editor-in-chief, until his death in 2009, in May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harvard held a dedicated to the 40th anniversary of the Journal of Differential Geometry. The journal is abstracted and indexed in MathSciNet, Zentralblatt MATH, Current Contents/Physical, Chemical & Earth Sciences, according to the Journal Citation Reports, the journal has a 2013 impact factor of 1.093. Official website Surveys in Differential Geometry web page
8.
Schwarz triangle
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In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. These can be defined generally as tessellations of the sphere. Each Schwarz triangle on a sphere defines a group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex, the value n/d means the vertex angle is d/n of the half-circle. When these are numbers, the triangle is called a Möbius triangle, and corresponds to a non-overlapping tiling. A Schwarz triangle is represented graphically by a triangular graph, each node represents an edge of the Schwarz triangle. Each edge is labeled by a value corresponding to the reflection order. Order-2 edges represent perpendicular mirrors that can be ignored in this diagram, the Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden. A Coxeter group can be used for a simpler notation, as for graphs, and = for. Density 10, The Schwarz triangle is the smallest hyperbolic Schwarz triangle and its triangle group is the triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the group, and all Hurwitz surfaces are tiled by the Schwarz triangle. The smallest Hurwitz group is the group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL. The triangle tiles the Bolza surface, a highly symmetric surface of genus 2, the triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in. A list of triangles with multiple noninteger angles is given in, 3D The general Schwarz triangle and the generalized incidence matrices of the corresponding polyhedra
9.
3-7 kisrhombille
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In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4,6, the image shows a Poincaré disk model projection of the hyperbolic plane. It is labeled V4.6.14 because each right triangle face has three types of vertices, one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the tessellation of the truncated triheptagonal tiling which has one square and one heptagon. The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a point to each rhombus. There are no mirror removal subgroups of, the only small index subgroup is the alternation, +. Three isohedral tilings can be constructed from this tiling by combining triangles, It is topologically related to a polyhedra sequence, see also the uniform tilings of the hyperbolic plane with symmetry. The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, just as the triangle group is a quotient of the modular group, the associated tiling is the quotient of the modular tiling, as depicted in the video at right. Hexakis triangular tiling Tilings of regular polygons List of uniform tilings Uniform tilings in hyperbolic plane
10.
Snub triheptagonal tiling
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In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex and it has Schläfli symbol of sr. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr, drawn in chiral pairs, with edges missing between black triangles, The dual tiling is called an order-7-3 floret pentagonal tiling, and is related to the floret pentagonal tiling. This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure and these figures and their duals have rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons, from a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Chapter 10, the Beauty of Geometry, Twelve Essays. Snub hexagonal tiling Floret pentagonal tiling Order-3 heptagonal tiling Tilings of regular polygons List of uniform planar tilings Kagome lattice Weisstein, Weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch