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Michael Frame is an American mathematician and retired Yale professor. [1] He is a co-author, along with Amelia Urry, of Fractal Worlds: Grown, Built, and Imagined. [2] At Yale, he was a colleague of Benoit Mandelbrot and helped Mandelbrot develop a curriculum within the mathematics department. [1]
Michael Frame was born in 1951 and grew up in St. Albans, West Virginia. [3] After leaving his physics major because the lab requirement was "something in biophysics with killing frogs," [3] Frame, a vegetarian, received a bachelor's degree in mathematics at Union College as a first-generation college student. [3] In 1978, he completed a PhD in mathematics at Tulane University.
Michael Frame came to work at Yale University at the invitation of his colleague Benoit Mandelbrot. At Yale, Frame called himself "the stupidest guy in the department...the dimmest bulb in the pack here," and focused on his teaching contributions. [4] He received the McCredie Prize for best use of technology in teaching at Yale College, the Dylan Hixon '88 Prize for teaching excellence in the natural sciences, and the Yale Phi Beta Kappa chapter's DeVane medal for undergraduate teaching. [1]
Benoit Mandelbrot includes a section on Michael Frame in his posthumously published autobiography The Fractalist: Memoir of a Scientific Maverick. [5] In the section, called "Michael Frame, Friend and Colleague," he calls Frame an "indispensable" professor. [5]
In 1997, Mandelbrot and Frame held a meeting of teachers of fractal geometry. [5] According to Mandelbrot, as far as he knew, this was the "first scientific meeting totally dedicated to the teaching of fractals. [5] This eventually culminated in the 2002 publication of the book Fractals, Graphics, and Mathematics Education, which was co-authored by Frame and Mandelbrot. [5] [6]
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.
A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the effective length, or increase the perimeter, of material that can receive or transmit electromagnetic radiation within a given total surface area or volume.
Gaston Maurice Julia was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related. He founded, independently with Pierre Fatou, the modern theory of holomorphic dynamics.
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.
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.
Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating program for personal computers.
"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in Science on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.
Forest Kenton Musgrave was a professor at The George Washington University in the USA. A computer artist who worked with fractal images, he worked on the Bryce landscape software and later as CEO/CTO of Pandromeda, Inc. developed and designed the innovative MojoWorld software.
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 mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic. By analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. These points are named after mathematician Michał Misiurewicz who first studied them.
The Beauty of Fractals is a 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathematics book became an unusual success.
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.
The Fractal Geometry of Nature is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.
The Lindy effect is a theorized phenomenon by which the future life expectancy of some non-perishable things, like a technology or an idea, is proportional to their current age. Thus, the Lindy effect proposes the longer a period something has survived to exist or be used in the present, the longer its remaining life expectancy. Longevity implies a resistance to change, obsolescence or competition and greater odds of continued existence into the future. Where the Lindy effect applies, mortality rate decreases with time. Mathematically, the Lindy effect corresponds to lifetimes following a Pareto probability distribution.
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.
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