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Fractal Art
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and Algorithmic composition, media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art. Fractal art (especially in the western world) is rarely drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics software, graphics programs are used to further modify the images produced. This is called post-processing. Non-fractal imagery may also be integrated into the artwork. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hindu Temple Design
Hindus (; ; also known as Sanātanīs) are people who religiously adhere to Hinduism, also known by its endonym Sanātana Dharma.Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pp. 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for people living in the Indian subcontinent. It is assumed that the term ''"Hindu"'' traces back to Avestan scripture Vendidad which refers to land of seven rivers as Avestan geography, Hapta Hendu which itself is a cognate to Sanskrit term ''Sapta Sindhuḥ''. (The term ''Sapta Sindhuḥ'' is mentioned in Rig Veda and refers to a North western Indian region of seven rivers and to India as a whole.) The Greek language, Greek cognates of the same terms are "''Indus''" (for the river) and "''India''" (for the land of the river). Likewise the Hebrew cognate ''hōd-dū'' refers to India mentioned in Hebrew BibleEsther 1:1. The term "''Hindu''" also implied a geographic, ethnic or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. Mathematics In mathematics, iteration may refer to the process of iterated function, iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nova Fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial \mathbb or transcendental function. It is the Julia set of the meromorphic function which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions , each of which is associated with a root of the polynomial, . In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point. Almost all points of the complex plane are associated with one of the roots of a given polynomial in the following way: the point is used as starting value for Newton's iteration , yielding a sequence of points If the sequence converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals. Origins As a biologist, Lindenmayer worked with yeast and filamentous fungi and studied the growth patterns of various types of bacteria, such as the cyanobacteria '' Anabaena ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flame Fractals
Fractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves' open-source code was later ported into Adobe After Effects graphics software and translated into the Apophysis fractal flame editor. Fractal flames differ from ordinary iterated function systems in three ways: * Nonlinear functions are iterated in addition to affine transforms. * Log-density display instead of linear or binary (a form of tone mapping) * Color by structure (i.e. by the recursive path taken) instead of monochrome or by density. The tone mapping and coloring are designed to display as much of the detail of the fractal as possible, which generally results in a more aesthetically pleasing image. Algorithm The algorithm consists of two steps: creating a histogram and then rendering the histogram. Creating the histogram First, one iterates a set of functions, starting from a randomly chosen point ''P = (P.x,P.y,P.c)'', where the third coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strange Attractors
In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an ''n''-dimensional Coordinate vector, vector. The attractor is a region in space (mathematics), ''n''-dimensional space. In Physics, physical systems, the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in Economics, economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented Geometry, geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function Systems
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński triangle. The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar fractal nature. Definition Formally, an iterated function system is a finite set of contraction mappings on a complete metric space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger Sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. Construction The construction of a Menger sponge can be described as follows: # Begin with a cube. # Divide every face of the cube into nine squares in a similar manner to a Rubik's Cube. This sub-divides the cube into 27 smaller cubes. # Remove the smaller cube in the middle of each face and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level 1 Menger sponge (resembling a void cube). # Repeat steps two and three for each of the remaining smaller cubes and continue to iterate ''ad infinitum''. The second iteration gives a level 2 sponge, the third iteration gives a level 3 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
