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Mathematics   Tags: mrs. cuza  

Last Updated: Aug 27, 2017 URL: http://carrollton.libguides.com/content.php?pid=688135 Print Guide RSS Updates
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Fractals

The Mandelbrot set is, for many, the quintessential fractal. When one zooms in on some part of the edge, one notices that the Mandelbrot set is, indeed, self-similar. Furthermore, if one zooms in even further on various sections of the edge, one obtains different Julia sets. In fact, it is "asymptotically similar to Julia sets near any point on its boundary," as proved in a theorem by the Chinese mathematician Tan Lei. [7]

Mandelbrot has managed not only to invent the discipline of fractal geometry, but has also popularized it through its applications to other areas of science. He clearly believed this was important, as he once stated [3]

The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.

As he hinted in How Long Is the Coast of Britain? fractal geometry comes in useful in representing natural phenomena; things such as coastlines, the silhouette of a tree, or the shape of snowflakes -- things are not easily represented using traditional Euclidean geometry. After all, no organic entity comes to mind when one contemplates a square or a circle. Equally, no simple shape from Euclidean geometry comes to mind when contemplating things such as the path of a river. Even the earth is not a perfect sphere, however convenient it may be for one's calculations to treat it as such. Furthermore, fractal geometry and chaos theory have important connections to physics, medicine, and the study of population dynamics. [7] However, even if the field lacked these links, it would be hard for those so inclined to resist the aesthetic appeal of most fractals.

Mandelbrot's non-traditional approach led him to invent an amazing and useful new form of mathematics. However, no mathematician can claim to have developed his results in complete isolation from anyone else's. Mandelbrot's discovery owes a great deal to the mathematicians who preceded him, such as Weierstrass and von Koch, but especially to JuliaFatou, andHausdorff. He also benefitted from access to computers, which allowed him not only to build upon the works of others in a new way -- one which had definitely not been done before -- but to use his preferred method of solving problems -- namely visualisation. Furthermore, his invention also makes a case for the importance of the study of pure mathematics: until Mandelbrot came along and united the eclectic ideas of Hausdorff,Julia, et al, they represented very abstract mathematical ideas from varying branches of (pure) mathematics. There is very little that would interest an ordinary biologist about set theory. However, through fractal geometry, many of these seemingly abstract ideas (from mathematicians who are relatively unknown outside of their own spheres of research) develop applications that other scientists and even non-scientists can appreciate. Thus, the work that eventually led to fractals and their applications are an excellent counterexample to the arguments of anyone who would dare to denigrate the study of pure mathematics.

For more information - Search:

MacTutor History of Mathematics Archive

      

    MATH

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    WHERE DOES MATH COME FROM?

    Online Journal Collections, Q&A Sites, & Brain Pop

    Links to Mathematics Publishers and Aggregator sites.

    History of Mathematics Websites

    History of Mathematics Websites

    These are websites that relate specifically to the historical development of mathematics. Some may not look so pretty but they are populated with important and relevant information.

    Electric Sheep

    Mandelbrot_sequence_new.gif (320×240)

    • Electric Sheep
      Electric Sheep is a distributed computing project for animating and evolving fractal flames, which are in turn distributed to the networked computers, which display them as a screensaver.
    • Electric Sheep

    Library of Congress -Call Numbers - Location of books in Founders Library

    LC Call No. & Subject Collection

    ONLINE CATALOG

    The online library catalog helps you find all library materials on the shelves in the library.

      

    CALL NUMBERS    SUBJECT GUIDE
    QA 8-QA 9 Mathematical Logic and Foundations
    QA 13-QA 29 History of Mathematics
    QA 76.76 Computer Science
    QA 184-QA 188 Linear and Multilinear Algebras and Matrix Theory
    QA 241-QA 246 Number Theory
    QA 269-QA 272 Game Theory, Economics, Social and Behavioral Sciences
    QA 273-QA 274 Probability Theory and Stochastic Processes
    QA 276 Statistics Computing and Software
    QA 299-QA 320 Real Functions Analysis
    QA 371-QA 377 Partial Differential Equations
    QA 402.5 Optimization
    QA 611 Topology
    QA 641-QA 649 Differential Geometry

                                                                         

        
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