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.
Vertex configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
3.
Wythoff symbol
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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 three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators, with a slight extension, Wythoffs symbol can be applied to all uniform polyhedra. However, the methods do not lead to all uniform tilings in euclidean or hyperbolic space. In three dimensions, Wythoffs construction begins by choosing a point on the triangle. If the distance of this point from each of the sides is non-zero, a perpendicular line is then dropped between the generator point and every face that it does not lie on. The three numbers in Wythoffs symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, the triangle is also represented with the same numbers, written. 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 if the corresponding mirror is active. The one impossible symbol | p q r implies the point is on all mirrors. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, the resulting figure has rotational symmetry only. The generator point can either be on or off each mirror and this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. A node is circled if the point is not on the mirror. There are seven generator points with each set of p, q, r, | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isnt Wythoff-constructible, There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the many such patterns in the hyperbolic plane are also listed. The list of Schwarz triangles includes rational numbers, and determine the set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a domain, colored by even. Selected tilings created by the Wythoff construction are given below, for a more complete list, including cases where r ≠2, see List of uniform polyhedra by Schwarz triangle
4.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also 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 classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. 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 often defined in terms of polar reciprocation about a concentric sphere. 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 reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, 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. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, 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 stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
5.
Heptagonal tiling
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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
6.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
7.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
8.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
9.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, 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, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to 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 often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, 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, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
10.
Regular hyperbolic tiling
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This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean, a Schläfli symbol describing an n-polytope equivalently describes a tessellation of a -sphere. Another related symbol is the Coxeter-Dynkin diagram which represents a group with no rings. For example, the cube has Schläfli symbol, and with its octahedral symmetry, the regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space, infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a scale. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and it cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane. A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and this table shows a summary of regular polytope counts by dimension. *1 if the number of dimensions is of the form 2k −1,2 if the number of dimensions is a power of two,0 otherwise, There are no Euclidean regular star tessellations in any number of dimensions. A one-dimensional polytope or 1-polytope is a line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol, norman Johnson calls it a ditel and gives it the Schläfli symbol. Although trivial as a polytope, it appears as the edges of polygons and it is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. The two-dimensional polytopes are called polygons, Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol, usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the forms, but connect in an alternate connectivity which passes around the circle more than once to complete. Star polygons should be called nonconvex rather than concave because the edges do not generate new vertices. The Schläfli symbol represents a regular p-gon, the regular digon can be considered to be a degenerate regular polygon
11.
Order-7 tetrahedral honeycomb
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In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has seven tetrahedra around each edge, all vertices are ultra-ideal with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement. It a part of a sequence of polychora and honeycombs with tetrahedral cells. In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation with Schläfli symbol and it has eight tetrahedra around each edge. All vertices are ultra-ideal with infinitely many tetrahedra existing around each vertex in an triangular tiling vertex arrangement. It has a construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram. In Coxeter notation the half symmetry is =, in the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation with Schläfli symbol. It has infinitely many tetrahedra around each edge, all vertices are ultra-ideal with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement. It has a construction as a uniform honeycomb, Schläfli symbol, Coxeter diagram, =. In Coxeter notation the half symmetry is =
12.
Hurwitz surface
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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
13.
(2,3,7) triangle group
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In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order,84, of its automorphism group. A note on terminology – the triangle group most often refers, not to the triangle group Δ, but rather to the ordinary triangle group D of orientation-preserving maps. Torsion-free normal subgroups of the group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface. To construct the triangle group, start with a triangle with angles π/2, π/3. This triangle, the smallest hyperbolic Schwarz triangle, tiles the plane by reflections in its sides, the triangle group is defined as the index 2 subgroup consisting of the orientation-preserving isometries, which is a Fuchsian group. It has a presentation in terms of a pair of generators, g2, g3, modulo the following relations, geometrically, these correspond to rotations by 2π/2, 2π/3, 2π/7 about the vertices of the Schwarz triangle. The triangle group admits a presentation in terms of the group of quaternions of norm 1 in an order in a quaternion algebra. More specifically, the group is the quotient of the group of quaternions by its center ±1. Then from the identity 3 =72 and we see that Q is a totally real cubic extension of Q. One chooses a suitable Hurwitz quaternion order Q H u r in the quaternion algebra, here the order Q H u r is generated by elements g 2 =1 η i j g 3 =12. In fact, the order is a free Z-module over the basis 1, g 2, g 3, g 2 g 3. Here the generators satisfy the relations g 22 = g 33 =7 = −1, extending the scalars from Q to R, one obtains an isomorphism between the quaternion algebra and the algebra M of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the triangle group as a specific Fuchsian group in SL and this can be visualized by the associated tilings, as depicted at right, the tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane. However, for many purposes, explicit isomorphisms are unnecessary, thus, traces of group elements can be calculated by means of the reduced trace in the quaternion algebra, and the formula tr =2 cosh
14.
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
15.
Hurwitz's automorphisms theorem
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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
16.
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
17.
PSL(2,7)
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In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements PSL is the second-smallest nonabelian simple group after the alternating group A5 on five letters with 60 elements, the general linear group GL consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. The subgroup SL consists of all matrices with unit determinant. Then PSL is defined to be the quotient group SL/ obtained by identifying I and −I, in this article, we let G denote any group isomorphic to PSL. G = PSL has 168 elements and this can be seen by counting the possible columns, there are 72−1 =48 possibilities for the first column, then 72−7 =42 possibilities for the second column. We must divide by 7−1 =6 to force the determinant equal to one and it is a general result that PSL is simple for n, q ≥2, unless = or. PSL is isomorphic to the symmetric group S3, and PSL is isomorphic to alternating group A4, in fact, PSL is the second smallest nonabelian simple group, after the alternating group A5 = PSL = PSL. The number of classes and irreducible representations is 6. The sizes of conjugacy classes are 1,21,42,56,24,24, the dimensions of irreducible representations 1,3,3,6,7,8. Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL, the order of group is 168=3*7*8, this implies existence of Sylows subgroups of orders 3,7 and 8. It is easy to describe the first two, they are cyclic, since any group of order is cyclic. Any element of conjugacy class 3A56 generates Sylow 3-subgroup, any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a group of order 8. It can be described as centralizer of any element from the conjugacy class 2A21, in the GL representation, a Sylow 2-subgroup consists of the upper triangular matrices. This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for p =2, however, PSL is also isomorphic to PSL, the special linear group of 3×3 matrices over the field with 2 elements. The Fano plane can be used to describe multiplication of octonions, the Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial x3y + y3z + z3x =0. It is a compact Riemann surface of genus g =3 and this bound is due to the Hurwitz automorphisms theorem, which holds for all g>1
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Immersion (mathematics)
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For a closed immersion in algebraic geometry, see closed immersion. In mathematics, an immersion is a function between differentiable manifolds whose derivative is everywhere injective. The function f itself need not be injective, only its derivative, a related concept is that of an embedding. A smooth embedding is an immersion f, M → N that is also a topological embedding. An immersion is precisely a local embedding – i. e. for any point x ∈ M there is a neighbourhood, U ⊂ M, of x such that f, U → N is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is taken to be the definition of an immersion. If M is compact, an immersion is an embedding. A regular homotopy is thus a homotopy through immersions, stephen Smale expressed the regular homotopy classes of immersions f, Mm → Rn as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence, morris Hirsch generalized Smales expression to a homotopy theory description of the regular homotopy classes of immersions of any m-dimensional manifold Mm in any n-dimensional manifold Nn. The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov, the primary obstruction to the existence of an immersion i, Mm → Rn is the stable normal bundle of M, as detected by its characteristic classes, notably its Stiefel–Whitney classes. Conversely, given such a bundle, an immersion of M with this bundle is equivalent to a codimension 0 immersion of the total space of this bundle. Thus, the dimension of the stable normal bundle, as detected by its highest non-vanishing characteristic class, is an obstruction to immersions. Since characteristic classes multiply under direct sum of vector bundles, this obstruction can be stated intrinsically in terms of the space M and its tangent bundle and this obstruction was stated by Whitney. For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 and this gave evidence to the Immersion Conjecture, namely that every n-manifold could be immersed in codimension n − α, i. e. in R2n−α. This conjecture was proven by Ralph Cohen, codimension 0 immersions are equivalently relative dimension 0 submersions, and are better thought of as submersions. A codimension 0 immersion of a manifold is precisely a covering map. By Ehresmanns theorem and Phillips theorem on submersions, a submersion of manifolds is a fiber bundle. Further, codimenson 0 immersions do not behave like other immersions, alternatively, this is by invariance of domain
19.
Small cubicuboctahedron
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In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces,48 edges, and 24 vertices and its vertex figure is a crossed quadrilateral. The cubicuboctahedron is a faceting of the rhombicuboctahedron and its name comes from that the square faces lying on the planes corresponding to the rhombic dodecahedron, has been replaced by six octagonal faces parallel to the square faces of the cube. It shares the vertex arrangement with the truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron, and with the small rhombihexahedron, the underlying polyhedron defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space, higher genus surfaces admit a metric of negative constant curvature, and the universal cover of the resulting Riemann surface is the hyperbolic plane. The corresponding tiling of the plane has vertex figure 3.8.4.8. If the surface is given the appropriate metric of curvature = −1 and this tiling may be denoted by the Wythoff symbol 34 |4, and is depicted at right. The corresponding tiling of the plane is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented to yield the Mathieu group M24, compound of five small cubicuboctahedra List of uniform polyhedra Eric W. Weisstein, Small cubicuboctahedron at MathWorld
20.
Vertex arrangement
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In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes, for example, a square vertex arrangement is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same vertex arrangement if they share the same 0-skeleton, the same set of vertices can be connected by edges in different ways. For example, the pentagon and pentagram have the same vertex arrangement, a vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a vertex arrangement. Infinite tilings can also share common vertex arrangements, for example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces. Polyhedra can also share an edge arrangement while differing in their faces, for example, of the ten nonconvex regular Schläfli-Hess polychora, there are only 7 unique face arrangements. Synonyms for special cases include company for a 2-regiment and army for a 0-regiment, n-skeleton - a set of elements of dimension n and lower in a higher polytope. Vertex figure - A local arrangement of faces in a polyhedron around a single vertex, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, archived from the original on 4 February 2007
21.
Order-7 heptagrammic tiling
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In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams. This tiling is a regular star-tiling, and has Schläfli symbol of, the heptagrams forming the tiling are of type. The overlapping heptagrams subdivide the hyperbolic plane into isosceles triangles,14 of which form each heptagram. Each point of the plane that does not lie on a heptagram edge belongs to the central heptagon of one heptagram. It has the vertex arrangement as the regular order-7 triangular tiling. The full set of edges coincide with the edges of a heptagonal tiling. The valance 6 vertices in this tiling are false-vertices in the one caused by crossed edges. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk
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Heptagrammic-order heptagonal tiling
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In geometry, the Heptagrammic-order heptagonal tiling is a regular star-tiling of the hyperbolic plane. The heptagonal faces overlap with density 3 and it has the same vertex arrangement as the regular order-7 triangular tiling. The full set of edges coincide with the edges of a heptagonal tiling. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk
23.
Dihedron
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A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons, a regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, the dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices. A dihedron can be considered a degenerate prism consisting of two n-sided polygons connected back-to-back, so that the object has no depth. The polygons must be congruent, but glued in such a way one is the mirror image of the other. This characterization holds also for the distances on the surface of a dihedron, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. The regular polyhedron is self-dual, and is both a hosohedron and a dihedron, in the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation, A regular ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol. It has two facets, which share all ridges, in common, polyhedron Polytope Weisstein, Eric W. Dihedron
24.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
25.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
26.
Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
27.
Triangular tiling
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In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the triangle is 60 degrees. The triangular tiling has Schläfli symbol of, Conway calls it a deltille, named from the triangular shape of the Greek letter delta. The triangular tiling can also be called a kishextille by a kis operation that adds a center point and it is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling, there are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors,111212 and 111112 from 121213 by combining 1 and 3, there is one class of Archimedean colorings,111112, which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The vertex arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb, the A*2 lattice can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing, the packing density is π⁄√12 or 90. 69%. Since the union of 3 A2 lattices is also an A2 lattice, the voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling, with identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color, the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid and these can be expanded to Platonic solids, five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbols. It is also related as a part of sequence of Catalan solids with face configuration Vn.6.6. Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling
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Order-8 triangular tiling
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In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of, having eight regular triangles around each vertex, the half symmetry = can be shown with alternating two colors of triangles, From symmetry, there are 15 small index subgroups by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, is the subgroup of. A larger subgroup is constructed, index 8, as with gyration points removed, the symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, from a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings. 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
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Infinite-order triangular tiling
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In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of. All vertices are ideal, located at infinity and seen on the boundary of the Poincaré hyperbolic disk projection, a lower symmetry form has alternating colors, and represented by cyclic symbol. The tiling also represents the fundamental domains of the *∞∞∞ symmetry and this tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol. 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. Weisstein, Eric W. Poincaré hyperbolic disk
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Wythoff construction
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoffs kaleidoscopic construction and it is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope, however, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point, if the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a point inside the spherical triangle enclosed by the mirrors. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron and this produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C and this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC and this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the corners by twice the angle at that point. Only even-numbered reflections of the vertex are used. The polyhedron has the Wythoff symbol |a b c, the process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian and they generally can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry, sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra, coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 HarEl, Z. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences,20 966–970, archived from the original on 4 February 2007. Displays Uniform Polyhedra using Wythoffs construction method Description of Wythoff Constructions Jenn, software that generates views of polyhedra and polychora from symmetry groups
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Uniform tilings in hyperbolic plane
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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
32.
Hyperbolic tiling
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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
33.
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
34.
Truncated heptagonal tiling
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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
35.
Triheptagonal tiling
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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
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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
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Rhombitriheptagonal tiling
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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
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Truncated triheptagonal tiling
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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
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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
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Order 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
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Euclidean tilings by convex regular polygons
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi and this means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons, There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for pair of vertices there is a symmetry operation mapping the first vertex to the second. Note that there are two mirror forms of 34.6 tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral, though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k -uniform or k -isogonal, if there are t orbits of tiles, as t -isohedral, if there are e orbits of edges, as e -isotoxal. K-uniform tilings with the vertex figures can be further identified by their wallpaper group symmetry. 1-uniform tilings include 3 regular tilings, and 8 semiregular ones, There are 20 2-uniform tilings,61 3-uniform tilings,151 4-uniform tilings,332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures. For edge-to-edge Euclidean tilings, the angles of the polygons meeting at a vertex must add to 360 degrees. A regular n -gon has internal angle 180 degrees, only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections. In particular, if three polygons meet at a vertex and one has an odd number of sides, the two polygons must be the same. If they are not, they would have to alternate around the first polygon, vertex types are listed for each. If two tilings share the two vertex types, they are given subscripts 1,2. There are 61 3-uniform tilings of the Euclidean plane,39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits. Chavey There are 151 4-uniform tilings of the Euclidean plane, Brian Galebachs search reproduced Krotenheerdts list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types
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John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and 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 problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and 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 especially 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, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, 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. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker