Welcome
to the Wonderful World of **Fractals**

This page will discuss the following topics: The history of fractals, the basic concept of fractals, frequently asked questions about fractals, and examples of some famous fractals. There will also be a list of links to some fractal web pages.

*"I coined fractal from
the Latin adjective fractus. The corresponding Latin verb
frangere means 'to break' to create irregular fragments. It is
therefore sensible - and how appropriate for our
need ! - that, in addition to "fragmented"
(as in fraction or refraction), fractus should also mean
"irregular", both meanings being preserved in
fragment."*** B. Mandelbrot**

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How long is the eastern coast of the US? There is
certainly a defined length for it. However, "fractally"
speaking, the length of the US eastern coast is *infinity*.
Why? A person given a map of the US can sit down with a ruler and
soon figure out the length. The problem is that repeating the
process with a larger scale map leads to a greater estimate of
length (Fig. 1). If the person actually went to the coast and
measured them directly, even greater estimates would result. This
fact shows that as the scale decreases, the estimated length
increases without limit. Therefore, if the scale of the
measurements were to be infinitely small, then the estimated
length would become infinitely large!!!

*Fig. 1 This picture is
showing that as the ruler gets smaller, the total length becomes
longer.*

Scale can be pictured in the mind as a measuring stick of a certain length. The finer the scale, the shorter the stick. Thus at any scale, think of a curve as being represented by a sequence of sticks (Fig. 1), joined end-to-end. Clearly, any feature shorter than the stick will not be shown from a map constructed this way. Of course, I don't think that anyone actually makes maps by laying sticks on the ground, but the stick analogy shows that the distortions are inevitably produced by the human eye, by the limited resolution of photographs, or even by the thickness of the pens used in drafting.

Mandelbrot proposed the idea of a fractal (short for
"fractional dimension") as a way to cope with problems
of scale in the real world. He defined a fractal to be any curve
or surface that is independent of scale. This property, referred
to as *self-similarity*, means that any portion of the
picture, if blown up in scale, would appear identical to the
large picture.

Fig. 2. Forming a cross by iteration of a simple procedure.

*This is an animated picture
of a simple fractal.*

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The first fractals were found during the 19th century. Cantor's dust created in 1872 is probably the most ancient known fractal. In 1890, Peano published his famous curve. Koch's curve was published in 1904. Then came Sierpinski's triangle in 1915. A few mathematicians noticed that there were more sophisticated means to define the dimension of an object. Fundamental work was done by Hausdorff (1919), then developed by Besicovitch (1935). The Hausdorff-Besicovitch dimension has played , later on, a major role in the domain of fractals.

Everything mentioned above was well known before Mandelbrot's
works, but these were scattered elements, and only understood by
a small number of specialists. Therefore little attention had
been paid to this knowledge and no one had thought of bringing
together all theses elements. Mandelbrot's brought the ideas
together and to develop an entirely new mathematical domain. The
term ** self-similar** seems to have appeared for the
first time in 1964, in an internal report at IBM (where
Mandelbrot was doing research) and in the title of a 1965 paper.
The acceptance of the word

Mandelbrot was then interested in noise on telecommunications
lines, in turbulence, in geophysical problems such as the length
of coastlines, in the hydrologic regime of streams that were
badly described with the theories known at that time. In all
these cases, he was capable of applying the same mathematical
approach and he found every time the notion of ** self-similarity**.

Peano Koch The Sierpinski's

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**Fractal World**

**Julia Set**

Web page created by **James Sun**

Last Updated: 03/12/98