Triangle Group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action. Definition Let ''l'', ''m'', ''n'' be integers greater than or equal to 2. A triangle group Δ(''l'',''m'',''n'') is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/''l'', π/''m'' and π/''n'' (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/''l'', 2π/''m'' and 2π/''n''. Therefore, if ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995). The radian is defined in the SI as being a dimensionless unit, with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing. Definition One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, \theta = \frac, where is the subtended angle in radians, is arc length, and is radius. A right angle is exactly \frac radians. The rotation angle (360°) corresponding to one complete revolution is the length of the circumference divided by the radius, which i ...
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Triangular Tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can ...
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Tetrakis Square Tiling
In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2. Conway, Burgiel, and Goodman-Strauss call it a kisquadrille, represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles. As a dual uniform tiling It is the dual tessellation of the truncated ...
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Bisected Hexagonal Tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''. Names Uniform colorings There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares. Related 2-uniform tilings The ''truncated trihexagonal tiling'' has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a central hexagon and surrounding triangles and square, in two different orientations. Circle packing The Truncated trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circ ...
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Tiling Regular 3-6 Triangular
Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester Theodor Tiling (1818–1871), physician and botanist *Reinhold Tiling (1893–1933), German rocket pioneer Other uses *Neuronal tiling *Tile drainage, an agriculture practice that removes excess water from soil *Tiling (crater), a small, undistinguished crater on the far side of the Moon See also *Brickwork *Packing (other) *Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...

Tiling Dual Semiregular V4-6-12 Bisected Hexagonal
Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester Theodor Tiling (1818–1871), physician and botanist *Reinhold Tiling (1893–1933), German rocket pioneer Other uses *Neuronal tiling *Tile drainage, an agriculture practice that removes excess water from soil *Tiling (crater), a small, undistinguished crater on the far side of the Moon See also *Brickwork *Packing (other) *Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...

Wallpaper Group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper. What this page calls pattern Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the bou ...
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Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the element of area of the surface, and is the line element along the boundary of . Here, is the Euler characteristic of . If the boundary is piecewise smooth, then ...
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Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard ...
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Group Presentation
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. S ...
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