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    Lesson 1 – Exploring the Mandelbrot set

Figure 1: Circle / Julia fractal

frac·tal (n) A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented solely by classical Euclidean geometry (the geometry of circles, squares, triangles, etc., we learned in high school). Fractals are used especially in computer modeling of irregular patterns and structures in nature. [French, from Latin fractus, past participle of frangere, to break.]

    The term "fractal" was coined by IBM Fellow and doctor of mathematics Benoit Mandelbrot, who expanded on ideas from earlier mathematicians and discovered similarities in chaotic and random events and shapes. Beginning in 1961, he published a series of studies on fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.

    In August 1985, Scientific American published an article on Mandelbrot's discoveries that introduced his ideas to the general public. Math and programming enthusiasts began writing their own programs to generate and explore the Mandelbrot set. In 1988, a group of programmers calling themselves the Stone Soup Group released the first version of FractInt – arguably the most popular freeware fractal generator of the next decade.

    In December 1997, Frederik Slijkerman began writing Ultra Fractal 2, incorporating ideas such as formula plug-ins, layering, and most recently – in UF 4 – animation capabilities. Ultra Fractal is incredibly powerful and versatile software that facilitates fractal exploration and provides the means to create amazingly artistic images.

    In 2003, Benoit Mandelbrot received the Japan Prize for his contributions to the study and understanding of complex systems – chaos and fractals. In his acceptance speech, Dr. Mandelbrot stated that in addition to the scientific and mathematical applications he is credited with revolutionizing, “…I have been granted the equally great privilege that work I had solely meant to be scientific also reached out to contribute both a new kind of art and a refreshed understanding of art as practiced since time immemorial.”

    While my own personal interest in fractals is definitely more artistically- than mathematically-based, I firmly believe that one must understand the tools and traditions of this art form. This first course introduces and discusses the nature of fractals, giving students a methodic and solid point of departure for further fractal exploration and artistic expression. Subsequent courses will delve more deeply into artistic techniques and topics.

    Let's begin! (Click on the link to Page 2, below.)

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