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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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The points of a Mandelbrot Set
are
bounded
as follows:
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.
| 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. 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. |
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| Koch's Snowflake: http://www.shodor.org/interactivate/activities/koch/index.html Sierpinski's
Triangle: A 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 Gray, S. B. (1992). Fractal Math. Journal
of Computers
in Mathematics |
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rating system for fractal pages: 
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