1.
Riemann surface
–
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the plane, locally near every point they look like patches of the complex plane. For example, they can look like a sphere or a torus or several sheets glued together, the main point of Riemann surfaces is that holomorphic functions may be defined between them. Every Riemann surface is a real analytic manifold, but it contains more structure which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface if, so the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not. Geometrical facts about Riemann surfaces are as nice as possible, and they provide the intuition and motivation for generalizations to other curves. The Riemann–Roch theorem is an example of this influence. There are several equivalent definitions of a Riemann surface, a Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas, the map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the maps between two overlapping charts are required to be holomorphic. A Riemann surface is a manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the supplement Riemann signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same, choosing an equivalence class of metrics on X is the additional datum of the conformal structure. A complex structure gives rise to a structure by choosing the standard Euclidean metric given on the complex plane. Showing that a structure determines a complex structure is more difficult. The complex plane C is the most basic Riemann surface, the map f = z defines a chart for C, and is an atlas for C. The map g = z* also defines a chart on C and is an atlas for C, the charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = is never compatible with A, in an analogous fashion, every non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way
2.
Hyperbolic geometry
–
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
3.
Hurwitz automorphisms theorem
–
A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, the theorem is named after Adolf Hurwitz, who proved it in. For example the double cover of the projective line y2 = xp −x branched at all points defined over the field has genus g=/2 but is acted on by the group SL2 of order p3−p. One of the themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvature K. It manifests itself in diverse situations and on several levels. While in the first two cases the surface X admits infinitely many automorphisms, a hyperbolic Riemann surface only admits a discrete set of automorphisms. By the uniformization theorem, any hyperbolic surface X – i. e. the Gaussian curvature of X is equal to one at every point – is covered by the hyperbolic plane. The conformal mappings of the surface correspond to orientation-preserving automorphisms of the hyperbolic plane, by the Gauss–Bonnet theorem, the area of the surface is A = − 2π χ = 4π. In order to make the automorphism group G of X as large as possible, we want the area of its fundamental domain D for this action to be as small as possible. If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a tiling of the plane, then p, q, and r are integers greater than one. Thus we are asking for integers which make the expression 1 − 1/p − 1/q − 1/r strictly positive and this minimal value is 1/42, and 1 − 1/2 − 1/3 − 1/7 = 1/42 gives a unique triple of such integers. This would indicate that the order |G| of the group is bounded by A/A ≤168. However, a more delicate reasoning shows that this is an overestimate by the factor of two, because the group G can contain orientation-reversing transformations, for the orientation-preserving conformal automorphisms the bound is 84. To obtain an example of a Hurwitz group, let us start with a -tiling of the hyperbolic plane and its full symmetry group is the full triangle group generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs, a Hurwitz surface is obtained by closing up a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus g. This will necessarily involve exactly 84 double triangle tiles, note that the polygons in the tiling are not fundamental domains – the tiling by triangles refines both of these and is not regular. Wythoff constructions yields further uniform tilings, yielding eight uniform tilings and these all descend to Hurwitz surfaces, yielding tilings of the surfaces. This is the last part of the theorem of Hurwitz, the smallest Hurwitz group is the projective special linear group PSL, of order 168, and the corresponding curve is the Klein quartic curve
4.
Schwarz triangle
–
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
5.
Hurwitz surface
–
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84 automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitzs theorem on automorphisms and they are also referred to as Hurwitz curves, interpreting them as complex algebraic curves. The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the triangle group, the finite quotient group is precisely the automorphism group. The group of complex automorphisms is a quotient of the triangle group. The Hurwitz surface of least genus is the Klein quartic of genus 3, with group the projective special linear group PSL, of order 84 =168 = 22·3·7. An interesting phenomenon occurs in the possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the automorphism group. The explanation for this phenomenon is arithmetic, namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet
6.
Klein quartic
–
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
7.
Fundamental domain
–
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 is a subset of the space which contains one point from each of these orbits. It serves as a 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 fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells, 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 nice set topologically. One typical condition is that D is almost an open set, in the sense that D is the difference of an open set in G with a set of measure zero. A fundamental domain always contains a regular set U, an open set moved around by G into disjoint copies. Frequently D is required to be a set of coset representatives with some repetitions. This is a situation in ergodic theory. If a fundamental domain is used to calculate an integral on X/G, for example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus. Examples in the three-dimensional Euclidean space R3. g, a parallelepiped, or a Wigner-Seitz cell, also called Voronoi cell/diagram. In the case of translational symmetry combined with other symmetries, the domain is part of the primitive cell. For example, for wallpaper groups the fundamental domain is a factor 1,2,3,4,6,8, the diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H. This famous diagram appears in all books on modular functions. Here, each region is a free regular set of the action of Γ on H. The boundaries are not a part of the regular sets
8.
3-7 kisrhombille
–
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
9.
Hyperbolic tiling
–
In hyperbolic geometry, a uniform hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex, for example 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, uniform tilings may be regular, quasi-regular or semi-regular. For right triangles, there are two regular tilings, represented by Schläfli symbol and, each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram,7 representing combinations of 3 active mirrors. An 8th represents an alternation operation, deleting alternate vertices from the highest form with all mirrors active, families with r =2 contain regular hyperbolic tilings, defined by a Coxeter group such as. Hyperbolic families with r =3 or higher are given by, hyperbolic triangles define compact uniform hyperbolic tilings. More symmetry families can be constructed from fundamental domains that are not triangles, selected families of uniform tilings are shown below. Each uniform tiling generates a dual tiling, with many of them also given below. There are infinitely many triangle group families and this article shows the regular tiling up to p, q =8, and uniform tilings in 12 families, and. The simplest set of hyperbolic tilings are regular tilings, which exist in a matrix with the regular polyhedra, the regular tiling has a dual tiling across the diagonal axis of the table. Self-dual tilings, etc. pass down the diagonal of the table, because all the elements are even, each uniform dual tiling one represents the fundamental domain of a reflective symmetry, *3333, *662, *3232, *443, *222222, *3222, and *642 respectively. As well, all 7 uniform tiling can be alternated, the triangle group, Coxeter group, orbifold contains these uniform tilings, The triangle group, Coxeter group, orbifold contains these uniform tilings. Because all the elements are even, each uniform dual tiling one represents the domain of a reflective symmetry, *4444, *882, *4242, *444, *22222222, *4222. As well, all 7 uniform tiling can be alternated, and this article shows uniform tilings in 9 families, and. The triangle group, Coxeter group, orbifold contains these uniform tilings, without right angles in the fundamental triangle, the Wythoff constructions are slightly different. For instance in the family, the snub form has six polygons around a vertex. In general the vertex figure of a tiling in a triangle is p.3. q.3. r.3
10.
Heptagonal tiling
–
In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of, having three regular heptagons around each vertex and this tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol. 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 symmetry group of the tiling is the group. The smallest Hurwitz surface is the Klein quartic, and the tiling has 24 heptagons. The dual order-7 triangular tiling has the symmetry group. Hexagonal tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes 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
11.
Truncated heptagonal tiling
–
In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two tetradecagons on each vertex and it has Schläfli symbol of t. The tiling has a configuration of 3.14.14. The dual tiling is called an order-7 triakis triangular tiling, seen as a triangular tiling with each triangle divided into three by a 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. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
12.
Triheptagonal tiling
–
In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex and it has Schläfli symbol of r. Compare to trihexagonal tiling with vertex configuration 3.6.3.6, drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms. The Beauty of Geometry, Twelve Essays, 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
13.
Truncated order-7 triangular tiling
–
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
14.
Order-7 triangular tiling
–
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of. The symmetry group of the tiling is the group. The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, the dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces. It is related to two star-tilings by the vertex arrangement, the order-7 heptagrammic tiling, and heptagrammic-order heptagonal tiling. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbol. 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. 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
15.
Rhombitriheptagonal tiling
–
In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, the tiling has Schläfli symbol rr. It can be seen as constructed as a rectified triheptagonal tiling, r, the dual tiling is called a deltoidal triheptagonal tiling, and consists of congruent kites. It is formed by overlaying an order-3 heptagonal tiling and a triangular tiling. 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. This tiling is related as a part of sequence of cantellated polyhedra with vertex figure. These vertex-transitive figures have reflectional symmetry, the Beauty of Geometry, Twelve Essays. 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
16.
Truncated triheptagonal tiling
–
In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon on each vertex and it has Schläfli symbol of tr. There is only one uniform coloring of a truncated triheptagonal tiling, each triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group. This tiling can be considered a member of a sequence of patterns with vertex figure. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. 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. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
17.
Snub triheptagonal tiling
–
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
18.
Order 3-7 kisrhombille
–
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
19.
SL2(R)
–
In mathematics, the special linear group SL or SL2 is the group of 2 ×2 real matrices with determinant one, SL =. It is a simple real Lie group with applications in geometry, topology, representation theory, SL acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL, more specifically, PSL = SL/, where I denotes the 2 ×2 identity matrix. It contains the modular group PSL, also closely related is the 2-fold covering group, Mp, a metaplectic group. Another related group is SL± the group of real 2 ×2 matrices with determinant ±1, SL is the group of all linear transformations of R2 that preserve oriented area. It is isomorphic to the symplectic group Sp and the special unitary group SU. It is also isomorphic to the group of unit-length coquaternions, the group SL± preserves unoriented area, it may reverse orientation. The quotient PSL has several interesting descriptions, It is the group of orientation-preserving projective transformations of the projective line R ∪. It is the group of automorphisms of the unit disc. It is the group of orientation-preserving isometries of the hyperbolic plane and it is the restricted Lorentz group of three-dimensional Minkowski space. Equivalently, it is isomorphic to the orthogonal group SO+. It follows that SL is isomorphic to the spin group Spin+, elements of the modular group PSL have additional interpretations, as do elements of the group SL, and these interpretations can also be viewed in light of the general theory of SL. Elements of PSL act on the projective line R ∪ as linear fractional transformations. This is analogous to the action of PSL on the Riemann sphere by Möbius transformations and it is the restriction of the action of PSL on the hyperbolic plane to the boundary at infinity. Elements of PSL act on the plane by Möbius transformations. This is precisely the set of Möbius transformations that preserve the upper half-plane and it follows that PSL is the group of conformal automorphisms of the upper half-plane. By the Riemann mapping theorem, it is also the group of automorphisms of the unit disc. The above formula can be used to define Möbius transformations of dual
20.
Modular group
–
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
21.
Systolic geometry
–
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
22.
Noam Elkies
–
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
23.
ArXiv
–
In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized
24.
Journal of Differential Geometry
–
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