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Dragon Curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently. Heighway dragon The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column ''Mathematical Games'' in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel '' Jurassic Park''. Construction The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dragon Curve
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Paperfolding Sequence
In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are: If a strip of paper is folded repeatedly in half in the same direction, i times, it will get 2^i-1 folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first 2^i-1 terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: Properties The value of any given term t_n in the regular paperfolding sequence, starting with n=1, can be found recursively as follows. Divide n by two, as many times as possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicorn Dragon Curve
The unicorn is a legendary creature that has been described since antiquity as a beast with a single large, pointed, spiraling horn projecting from its forehead. In European literature and art, the unicorn has for the last thousand years or so been depicted as a white horse-like or goat-like animal with a long straight horn with spiralling grooves, cloven hooves, and sometimes a goat's beard. In the Middle Ages and Renaissance, it was commonly described as an extremely wild woodland creature, a symbol of purity and grace, which could be captured only by a virgin. In encyclopedias, its horn was described as having the power to render poisoned water potable and to heal sickness. In medieval and Renaissance times, the tusk of the narwhal was sometimes sold as a unicorn horn. A bovine type of unicorn is thought by some scholars to have been depicted in seals of the Bronze Age Indus Valley civilization, the interpretation remaining controversial. An equine form of the unicor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy C Curve
In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Paul Lévy, who was the first to describe its self-similarity properties as well as to provide a geometrical construction showing it as a representative curve in the same class as the Koch curve. It is a special case of a period-doubling curve, a de Rham curve. L-system construction If using a Lindenmayer system then the construction of the C curve starts with a straight line. An isosceles triangle with angles of 45°, 90° and 45° is built using this line as its hypotenuse. The original line is then replaced by the other two sides of this triangle. At the second stage, the two new lines each form the base for another right-angled isosceles triangle, and are replaced by the other two sides of their respective triangle. So, after two stag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Developing Terdragon Curve
Development or developing may refer to: Arts *Development hell, when a project is stuck in development *Filmmaking, development phase, including finance and budgeting *Development (music), the process thematic material is reshaped *Photographic development * ''Development'' (album), a 2002 album by Nonpoint Business *Business development, a process of growing a business *Career development *Corporate development, a position in a business *Energy development, activities concentrated on obtaining energy from natural resources *Green development, a real estate concept that considers social and environmental impact of development *Land development, altering the landscape in any number of ways * Land development bank, a kind of bank in India *Leadership development *New product development *Organization development *Professional development *Real estate development *Research and development *Training and development *Fundraising, also called "development" Biology and medicine * Chi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex-base System
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary number, imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965W. Penney, A "binary" system for complex numbers, JACM 12 (1965) 247-248.). In general Let D be an integral domain \subset \C, and , \cdot, the Absolute value (algebra)#Types of absolute value, (Archimedean) absolute value on it. A number X\in D in a positional number system is represented as an expansion : X = \pm \sum_^ x_\nu \rho^\nu, where : The cardinality R:=, Z, is called the ''level of decomposition''. A positional number system or coding system is a pair : \left\langle \rho, Z \right\rangle with radix \rho and set of digits Z, and we write the standard set of digits with R digits as : Z_R := \. Desirable are coding systems with the features: * Every number in D, e. g. the integers \Z, the Gaussian integers \Z[\mathrm i] or the integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twindragon
A dragon curve is any member of a family of Self-similarity, self-similar fractal curves, which can be approximated by recursion, recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently. Heighway dragon The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column ''Mathematical Games'' in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel ''Jurassic Park (novel), Jurassic Park''. Construction The Heighway dragon can be constructed from a base line segment by repeated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-filling Curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. Definition Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a ''curve'': In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
