Dietmar Saupe (born 1954) is a fractal researcher and professor of computer science, Department of Computer and Information Science, University of Konstanz, Germany.
Saupe's book, Chaos and Fractals, won the Association of American Publishers award for Best Mathematics Book of the Year in 1992. His current research interests include computer graphics, scientific visualization, and image processing.
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Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.
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.
The Mandelbrot set is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at , i.e., for which the sequence , , etc., remains bounded in absolute value.
In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and 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 compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image. Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.
Clifford Alan Pickover is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research Center in Yorktown, New York, where he was editor-in-chief of the IBM Journal of Research and Development. He has been granted more than 700 U.S. patents, is an elected Fellow for the Committee for Skeptical Inquiry, and is author of more than 50 books, translated into more than a dozen languages.
Scott Draves is the inventor of fractal flames and the leader of the distributed computing project Electric Sheep. He also invented patch-based texture synthesis and published the first implementation of this class of algorithms. He is also a video artist and accomplished VJ.
Heinz-Otto Peitgen is a German mathematician and was President of Jacobs University from January 1, 2013 to December 31, 2013. Peitgen contributed to the study of fractals, chaos theory, and medical image computing, as well as helping to introduce fractals to the broader public.
Joseph Yehuda Halpern is an Israeli-American professor of computer science at Cornell University. Most of his research is on reasoning about knowledge and uncertainty.
The diamond-square algorithm is a method for generating heightmaps for computer graphics. It is a slightly better algorithm than the three-dimensional implementation of the midpoint displacement algorithm, which produces two-dimensional landscapes. It is also known as the random midpoint displacement fractal, the cloud fractal or the plasma fractal, because of the plasma effect produced when applied.
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot.
In dynamical systems theory, the Gingerbreadman map is a chaotic two-dimensional map. It is given by the piecewise linear transformation:
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are available to play or tinker with fractals. Some programmers create fractal software for themselves because of the novelty and because of the challenge in understanding the related mathematics. The generation of fractals has led to some very large problems for pure mathematics.
Midhat Gazalé was an international telecommunications and space consultant and a visiting Professor of Telecommunications and Computer Management at the University of Paris IX. He served as President of AT&T-France, as chairman of the board for Sperry-France and for International Computers-France, and as an executive and research scientist for other major companies. He was made Chevalier dans l'Ordre national du Mérite in 1981. He was born in Alexandria and was a special advisor to the Egyptian prime minister for science and technology.
The Fractal Geometry of Nature is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.
The Hochschule Konstanz, is a German university located in Konstanz, Baden-Württemberg, Germany, in southern Germany close to the border with Switzerland. The university is a member of Internationale Bodensee-Hochschule.
Michael Batty is a British academic currently appointed as Bartlett Professor of Planning in The Bartlett at University College London. His work spans the fields of urban planning, geography and spatial data science. He has been Director—now Chairman—of the Centre for Advanced Spatial Analysis, set up when he was appointed to UCL in 1995. His research and the work of CASA is focused on computer models of city systems. He was awarded the William Alonso Prize of the Regional Science Association in 2011 for his book Cities and Complexity, the same prize a second time for his book The New Science of Cities in 2017–2018, the University Consortium GIS Research Award in 2012, and the Lauréat Prix International de Géographie Vautrin Lud, the so-called 'Nobel for geography', in 2013. In 2015, he was awarded the Founder's Medal of the Royal Geographical Society and in 2016, the Gold Medal of the Royal Town Planning Institute (RTPI). He also received the Senior Scholar Award of the Complex Systems Society in September 2016.
Dietmar is a German forename.
Robert Luke Devaney is an American mathematician who is the Feld Family Professor of Teaching Excellence at Boston University. His research involves dynamical systems and fractals.
