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Burning Ship Fractal
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function: :z_ = (, \operatorname \left(z_n\right), +i, \operatorname \left(z_n\right), )^2 + c, \quad z_0=0 in the complex plane \mathbb which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations. Virtually all images of the Burning Ship fractal are reflected vertically for aesthetic purposes, and some are also reflected horizontally. Implementation The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code. for each pixel (''x'', ''y'') on the screen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto E
Otto is a masculine German given name and a surname. It originates as an Old High German short form (variants '' Audo'', '' Odo'', '' Udo'') of Germanic names beginning in ''aud-'', an element meaning "wealth, prosperity". The name is recorded from the 7th century ( Odo, son of Uro, courtier of Sigebert III). It was the name of three 10th-century German kings, the first of whom was Otto I the Great, the first Holy Roman Emperor, founder of the Ottonian dynasty. The Gothic form of the prefix was ''auda-'' (as in e.g. '' Audaþius''), the Anglo-Saxon form was ''ead-'' (as in e.g. '' Eadmund''), and the Old Norse form was '' auð-''. Due to Otto von Bismarck, the given name ''Otto'' was strongly associated with the German Empire in the later 19th century. It was comparatively frequently given in the United States (presumably in German American families) during the 1880s to 1890s, remaining in the top 100 most popular masculine given names in the US throughout 1880–1898, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vector (geometry), vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or ' of the product is the product of the two absolute values, or moduli, and the angle or ' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol , which can be sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbrot Set
The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting at z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an infinitely complicated Boundary (topology), boundary that reveals progressively ever-finer Recursion, recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Riemann Equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable. These equations are and where and are real differentiable function#Differentiability in higher dimensions, bivariate differentiable functions. Typically, and are respectively the real part, real and imaginary parts of a complex number, complex-valued function of a single complex variable where and are real variables; and are real differentiable functions of the real variables. Then is complex differentiable at a complex point if and only if the partial derivatives of and satisfy the Cauchy–Riemann equations at that point. A holomorphic function is a complex function that is differentiable at eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford A
Clifford may refer to: People * Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford * Baron Clifford *Baron Clifford of Chudleigh *Baron de Clifford * Clifford baronets * Clifford family (bankers) * Jaryd Clifford * Justice Clifford (other) * Lord Clifford (other) Arts, entertainment, and media *''Clifford the Big Red Dog'', a series of children's books ** Clifford (character), the central character of ''Clifford the Big Red Dog'' ** ''Clifford the Big Red Dog'' (2000 TV series), 2000 animated TV series **'' Clifford's Puppy Days'', 2003 animated TV series **'' Clifford's Really Big Movie'', 2004 animated movie ** ''Clifford the Big Red Dog'' (2019 TV series), 2019 animated TV series ** ''Clifford the Big Red Dog'' (film), 2021 live-action movie * ''Clifford'' (film), a 1994 film directed by Paul Flaherty * Clifford (Muppet) Mathematics *Clifford algebra, a type of associative algebra, named after ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a Shape, 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 geometry, 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 Scaling (geometry), 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). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burning Ship Fractal Zoom-out 64
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion does not always result in fire, because a flame is only visible when substances undergoing combustion vaporize, but when it does, a flame is a characteristic indicator of the reaction. While activation energy must be supplied to initiate combustion (e.g., using a lit match to light a fire), the heat from a flame may provide enough energy to make the reaction self-sustaining. The study of combustion is known as combustion science. Combustion is often a complicated sequence of elementary radical reactions. Solid fuels, such as wood and coal, first undergo endothermic pyrolysis to produce gaseous fuels whose combustion then supplies the heat required to produce more of them. Combustion is often hot enough that incandescent light in the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudocode
In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actions and conditions. Although pseudocode shares features with regular programming languages, it is intended for human reading rather than machine control. Pseudocode typically omits details that are essential for machine implementation of the algorithm, meaning that pseudocode can only be verified by hand. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation. The reasons for using pseudocode are that it is easier for people to understand than conventional programming language code and that it is an efficient and environment-independent description of the key principles of an algorithm. It is commonly used in textbooks and scientific publications to document ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demoscene
The demoscene () is an international computer art subculture focused on producing demos: self-contained, sometimes extremely small, computer programs that produce audiovisual presentations. The purpose of a demo is to show off computer programming, programming, visual art, and musical skills. Demos and other demoscene productions (graphics, music, videos, games) are shared, voted on and released online at festivals known as Demoscene#Parties, demoparties. The scene started with the home computer revolution of the early 1980s, and the subsequent advent of software cracking. Crackers altered the code of computer games to remove copy protection, claiming credit by adding introduction screens of their own ("crack intro, cracktros"). They soon started competing for the best visual presentation of these additions. Through the making of intros and stand-alone demos, a new community eventually evolved, independent of the gaming and Warez scene, software sharing scenes. Demos are informa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nebulabrot
The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark ('' tika''), a traditional oval crown (''ushnisha''), and ringlet of hair. Discovery The ''Buddhabrot'' rendering technique was discovered by Melinda Green, who later described it in a 1993 Usenet post to sci.fractals. Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988, Linas Vepstas relayed similar images to Cliff Pickover for inclusion in Pickover's then-forthcoming book ''Computers, Pattern, Chaos, and Beauty''. This led directly to the discovery of Pickover stalks. Noel Griffin also implemented this idea in the 1993 "Mandelcloud" option in the Fractint renderer. However, these researchers did not filter out non-escaping trajectories required to produce the ghostly forms reminiscent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
