Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences. In addition, the quotient difference becomes arbitrarily large as the summation index increases.

Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.

Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals. One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake.

Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet.

That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations.

The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975.

That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations.

In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension.

Freeman and Co., 1982.

New York: Springer-Verlag, 1988.

. Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991.

New York: Springer-Verlag, 1992.

Boston: Academic Press Professional, 1993.

Jones, Jesse; Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993.

"This book has been written for a wide audience..." Includes sample BASIC programs in an appendix. Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; Exploring Fractals on the Macintosh, Addison Wesley, 1995.

It never occurred to them that the Africans might have been using a form of mathematics that they hadn’t even discovered yet." In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a.

Mandelbrot); Masson, 1996.

, and New York: Springer-Verlag, 1996.

John Wiley and Sons, 1997.

Clarke documentary introduction to the fractal concept and the Mandelbrot set.) Liu, Huajie; Fractal Art, Changsha: Hunan Science and Technology Press, 1997, . Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B.

The current understanding is that fractals are ubiquitous in cell biology, from proteins, to organelles, to whole cells. ==In creative works== Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of Jackson Pollock's (1912–1956) paintings which were created by pouring paint directly onto his horizontal canvases Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.

Available in PDF version at. ==External links== Scaling and Fractals presented by Shlomo Havlin, Bar-Ilan University Hunting the Hidden Dimension, PBS NOVA, first aired August 24, 2011 Benoit Mandelbrot: Fractals and the Art of Roughness, TED, February 2010 Technical Library on Fractals for controlling fluid Equations of self-similar fractal measure based on the fractional-order calculus（2007） Mathematical structures Topology Computational fields of study

Available in PDF version at. ==External links== Scaling and Fractals presented by Shlomo Havlin, Bar-Ilan University Hunting the Hidden Dimension, PBS NOVA, first aired August 24, 2011 Benoit Mandelbrot: Fractals and the Art of Roughness, TED, February 2010 Technical Library on Fractals for controlling fluid Equations of self-similar fractal measure based on the fractional-order calculus（2007） Mathematical structures Topology Computational fields of study

Bielefeld: Transcript, 2014.

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