Orbifold notation

In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, their analogues on the hyperbolic plane; the following types of Euclidean transformation can occur in a group described by orbifold notation: reflection through a line translation by a vector rotation of finite order around a point infinite rotation around a line in 3-space glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.

Each group is denoted in orbifold notation by a finite string made up from the following symbols: positive integers 1, 2, 3, … the infinity symbol, ∞ the asterisk, * the symbol o, called a wonder and a handle because it topologically represents a torus closed surface. Patterns repeat by two translation; the symbol ×, called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, assumed to contain two independent translations; each symbol corresponds to a distinct transformation: an integer n to the left of an asterisk indicates a rotation of order n around a gyration point an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line an × indicates a glide reflection the symbol ∞ indicates infinite rotational symmetry around a line.

By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way; the exceptional symbol o indicates that there are two linearly independent translations. An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, p≠q. An object is chiral; the corresponding orbifold is non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol; each feature has a value: n without or before an asterisk counts as n − 1 n n after an asterisk counts as n − 1 2 n asterisk and × count as 1 o counts as 2. Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e. the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are those with the sum of the feature values equal to 2.

Otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic: 1* and *11 22 and 221 *22 and *221 2* and 2*1; this is. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side, thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. A 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.

*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, 2-fold gyration points as small green squares. A first few hyperbolic groups, ordered by their Euler characteristic are: Mutation of orbifolds Fibrifold notation - an extension of orbifold notation for 3d space groups John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, W

Wythoff symbol

In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.

A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.

The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...

The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con

John Horton Conway

John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.

Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.

There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.

In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.

He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.

Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.

R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.

The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.

When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form

Hyperbolic geometry

In mathematics, hyperbolic 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. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is 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.

Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.

Two intersecting lines have the same 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, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry 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 distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.

These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.

The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though