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Orbifold Notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups in three dimensions, point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic geometry, hyperbolic plane (H^2). Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientation (geometry), orientations), in contrast to rotation around a fixed axis, rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Groups In One Dimension
A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function ''f''(''x'') for, say, the color at position ''x''. The only nontrivial point group in 1D is a simple reflection. It can be represented by the simplest Coxeter group, A1, or Coxeter-Dynkin diagram . Affine symmetry groups represent translation. Isometries which leave the function unchanged are translations ''x'' + ''a'' with ''a'' such that ''f''(''x'' + ''a'') = ''f''(''x'') and reflections ''a'' − ''x'' with a such that ''f''(''a'' − ''x'') = ''f''(''x''). The reflections can be represented by the affine Coxeter group infin; or Coxeter-Dynkin diagram representing two reflections, and the translational symmetry as infin;sup>+, or Coxeter-Dynkin diagram as the composite of two reflections. Point group For a pattern without translational symmetry there are the following possibilities (1D point groups): * the sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One Dimension
A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number. Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve. In physical space, a 1D subspace is called a "linear dimension" ( rectilinear or curvilinear), with units of length (e.g., metre). In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field K is a one-dimensional vector space over itself. The projective line over K, denoted \mathbf P^1(K), is a one-dimensional space. In particular, if the field is the complex numbers \mathbb, then the complex projective line \mathbf P^1(\mathbb) is one-dimensional with respect to \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digon
In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. It may also be viewed as a representation of a graph theory, graph with two vertices, see "Generalized polygon". A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol . It may be constructed on a spherical geometry, sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a spherical lune, lune. The digon is the simplest abstract polytope of rank 2. A truncation (geometry), truncated ''digon'', t is a square, . An Alternation (geometry), alternated digon, h is a monogon, . In different fields In Euclidean geometry The digon can have one of two visual representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bullet (typography)
In typography, a bullet or bullet point, , is a typographical symbol or glyph used to introduce items in a list. For example: * Monica * Erica * Rita * Tina * Sandra * Mary * Jessica The bullet symbol may take any of a variety of shapes, such as circular, square, diamond or arrow. Typical word processor software offers a wide selection of shapes and colors. Several regular symbols, such as (asterisk), (hyphen), ( period), and even (lowercase Latin letter O), are conventionally used in ASCII-only text or other environments where bullet characters are not available. Historically, the index symbol (representing a hand with a pointing index finger) was popular for similar uses. Lists made with bullets are called bulleted lists. The HTML element name for a bulleted list is " unordered list", because the list items are not arranged in numerical order (as they would be in a numbered list). "Bullet points" Items—known as "bullet points"—may be short phrases, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two Dimension
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Mathematics The number 2 is the second natural number after 1. Each natural number, including 2, is constructed by succession, that is, by adding 1 to the previous natural number. 2 is the smallest and the only even prime number, and the first Ramanujan prime. It is also the first superior highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base 10, all multiples of 2 will end in 0, 2, 4, 6, or 8; more generally, in any even base, even numbers will end with an even digit. A digon is a polygon with two sides (or edges) and two vertices. Two distinct points in a plane are always sufficient to define a unique line in a nontri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant under some Transformation (function), transformations, such as Translation (geometry), translation, Reflection (mathematics), reflection, Rotation (mathematics), rotation, or Scaling (geometry), scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a space, spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including scientific model, theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag Of Hong Kong
The Regional Flag of the Hong Kong Special Administrative Region of the People's Republic of China depicts a white stylised five-petal Hong Kong orchid tree (''Bauhinia × blakeana'') flower in the centre of a Chinese red field. Its design is enshrined in Hong Kong's Basic Law, the territory's constitutional document, and regulations regarding its use, prohibition of use, desecration, and manufacture are stated in the Regional Flag and Regional Emblem Ordinance. The original design of the flag of Hong Kong was unveiled on 4 April 1990 at the Third Session of the Seventh National People's Congress. The current design was approved on 10 August 1996 at the Fourth Plenum of the Preparatory Committee of the Hong Kong Special Administrative Region. The use of the flag is regulated by laws passed by the 58th executive meeting of the State Council held in Beijing. The flag was officially adopted and hoisted on 1 July 1997, during the handover ceremony marking the handover from t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snowflake
A snowflake is a single ice crystal that is large enough to fall through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. 1, pp. 100–107.Hobbs, P.V. 1974. Ice Physics. Oxford: Clarendon Press. Snow appears white in color despite being made of clear ice. This is because the many small crystal facets of the snowflakes scatter the sunlight between them. Each flake begins by forming around a tiny particle, called its nucleus, accumulating water droplets, which freeze and slowly form a crystal. Complex shapes emerge as the flake moves through differing temperature and humidity zones in the atmosphere, and possibly combines with other snowflakes. Because of this, snowflakes tend to look very different from one another. However, they may be categorized in eight broad classifications and at least 80 individual variants. The main constituent shapes for ice crystals, from which combinations may occur, are ''needle'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
