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The Mandelbrot Set weds the graphing of complex numbers to the recursive power of modern computers.
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MandelZoom (C) Louis P. Santillan 2001-2002


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For source code, email Louis here.

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More fractals . . . 

slide show of fractals

The points of a Mandelbrot Set are bounded as follows:
Mandelbrot Set graph

Domain:  { x: -2  < x < +2 }

Range: {yi: -2i < x < +2i}

Size: radius or distance from (0,0) < 2.

The full Mandelbrot Set is plotted within the inscribed circle  of radius < 2.  Other views showing the fractal edge are displayed by zooming in on only a portion of the bounded area.

     Sample calculation:
Sample calculation

Mathematicians in the early 20th century investigated curves that had highly intricate and detailed shapes.  Moreover, they realized that while a region might be bounded and thus the area finite, the perimeter or border might seem to be infinite.  These curves - the Koch Snowflake for example - with finite area and infinite perimeter, were given the name of "pathological."  This particular area of research in mathematics has generated colorful names: Cantor's dust, Polya's sweeps, Peano's dragons, Sierpinski's carpet and others.

When the edge of a curve under many iterations is broken, repeated, scaled down, and then scaled down again as the iterations progress, the curve has now become known as a fractal.  This relatively new word in mathematics was first coined by Benoit B. Mandelbrot and introduced to mathematicians and computer scientists in The Fractal Geometry of Nature published in 1983.  He suggested the coastline of Great Britain, with its capes within capes, and bays within bays, surrounding a known land mass, be the mental model.  Librarians noted the August, 1985 article in Scientific American on the "new fractal geometry" was extremely popular.Cover Scientific American

Other writers have suggested the viewer look at a curve and then zoom in for a closer look.  Like Sherlock Holmes, with a magnifying glass, the viewer must search.  He does not know where on the curve he views.  Curves within curves.  Patterns within patterns.  Iterations within iterations.......The same repetitious but scaled pattern appear in a true fractal.  The area is bounded, but the perimeter seems to be infinite.

Technology made this formerly tedious mathematics possible.  With the help of the computer in the 1960s, Mandelbrot returned to earlier research questions first posed between 1915 and 1930 by French mathematicians Gaston Julia and Pierre Fatou: "Does the iteration of a function drive the result off to infinity or does the result approach and remain close to some well-defined value?"  With recursion no longer difficult, a mathematician could write a program and then simply wait, with great curiosity, for the speed of the computer to provide some answers.

In addition to Mandelbrot, John Hubbard of Cornell University and Adrian Douady of the Ecole Normale Superieure, Paris, and P. T. Bierberg of Finland are credited with acute observations, research, and early publications.

Great Fractal Links

Koch's Snowflake:

Sierpinski's Triangle:

rating system for fractal pages:

For Middle and High School students:

A gallery:

NCSA Site:

Power Point slides from Stanford:

Fascinating graphics:

A Pythagorean Tree Fractal

From a great technical university in Sweden:

and other early printed sources
Heppenheimer, T. A. (1985).  Mathematics at the receiving end. Mosaic.
National Science Foundation, 16 (4), 37-47.

Gray, S. B. (1992).  Fractal Math.  Journal of Computers in Mathematics
and Science Teaching.  11 (1), 31-38.

Some important concepts and the basic research question:
Once the distance from the origin exceeds 2, it never returns to less than 2.  Some complex numbers take quite a number of iterations to reach this size.  Others never do, and still others pass 2 almost immediately. The set of values that does NOT escape to infinity, and that remains bounded under many iterations of a function, is called the Julia set of that function to honor the French mathematician who first made these painstaking investigations in the early 20th century. Quote
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