FESTIVALS & FLAMES
The Mandelbrot Set
Fractals can also be created by repeatedly
calculating a simple equation over and over
thousands or millions of times. It was doing this
that led to the discovery of the Mandelbrot set.
It was in 1975 that Benoit Mandelbrot (19242010) coined the term “fractal” whilst working
on the concept of self-similarity. He worked at
the computer company IBM and therefore used
their computers to perform iterations quickly,
initially looking at self-similarity in a range of
contexts from the stock market to the way stars
spread out across the universe. In the early
1980s he investigated the fractal now known
as the Mandelbrot set. The equation he devised
is relatively simple: zn+1=zn2+c. You get the
next number in the sequence zn+1 by taking the
a
current number zn, squaring it and then adding
another number, c, to it. Then repeat. Choose a
number for c and while some values of c cause
z to get larger and larger, others become a
stable value or keep moving around without ever
getting larger than 2. If you plot all the points
where c never leads to a value larger than 2,
you get image A, then if you shade the unstable
points of the graphs indicating how many
iterations it took them to become larger than
two (more iterations = brighter) you end up with
image B. The amazing thing about this fractal is
that you can zoom into it for ever, (image C) and
the maths involved is helping examine things as
far apart as the human heartbeat and drawing
animation.
b
c
Image: Wolfgang Beyer
FOR MORE INFORMATION ABOUT THE FASCINATING WORLD OF FRACTALS
read http://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf
visit http://fractalfoundation.org
or see the video programme:
Fractals – Hunting the Hidden Dimension
https://www.youtube.com/watch?v=s65DSz78jW4
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