What is a Fractal?

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What is a Fractal?

• A fractal is a mathematical object that is both
  self-similar and chaotic.
• self-similar As you magnify, you see the object
  over and over again in its parts.
• chaotic Fractals are infinitely complex.
• Amazingly, these beautiful objects of
  breath-taking complexity are generated by
  relatively simple mathematical processes.

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The first fractals were discovered by a french
Mathematician named Gaston Julia who discovered
them decades before the advent of computer
graphics.
Julias work was rediscovered by Benoit
Mandelbrot. The most famous of all fractals is
the Mandelbrot set.
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...And we can continue to zoom in. As we
magnify the object, we see the same thing over
and over again.....This is Self Similarity
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These two pictures are interesting because they
show the same portion of the mandelbrot set
colored differently.

• The choice of color scheme really influences what
  we see in the picture.
• Is this mathematics or art?

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The Fractal Geometry of Nature
Why is geometry often described as cold and dry?
One reason lies in its inability to describe the
shape of a cloud, a mountain, a coastline, or a
tree. Clouds are not spheres, mountains are not
cones, coastlines are not circles, and bark is
not smooth, nor does lightning travel in a
straight line... ...Nature exhibits not simply a
higher degree but an altogether different level
of complexity. The number of distinct scales of
length of patterns is for all purposes
infinite. The existence of these patterns
challenges us to study those forms that Euclid
leaves aside as being formless, to investigate
the morphology of the amorphous. Mathematicians
have disdained this challenge, however, and have
increasingly chosen to flee from nature by
devising theories unrelated to anything we can
see or feel. ---Benoit Mandelbrot (1984)
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Aristid Lindenmeyer
Aristid Lindenmeyer invented L-Systems to model
plant growth. See the fractal he is holding?
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Fractal Landscapes

• In the following slides we will see some
  landscapes that are progressively more complex.
  These landscapes are NOT drawings. They are
  created entirely by a computer using fractal
  interpolation.
• This is the procedure that is used by special
  effects artists to create computer generated
  scenes for the big screen. (Independence Day
  was full of them.) Of course, mine are much
  cruder than theirs. They were produced in about
  an hour on my (ancient) pentium computer at home
  by a program called VISTAPRO.

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This is the crudest version. Nature made of
triangles.
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In this picture, the grid of triangles is a bit
finer
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The triangles are finer still and the mountains
in the distance begin to look a bit better.
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The next step. The mountains in the background
look like real mountains.
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Now we add some texture to the triangles.
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Here we add some crude clouds and improve the
landscape.
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Here we add some crude trees, and improve the
landscape and the clouds.
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This is the best resolution I got.
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We can make some artistic choices of color to
give us the same landscape in different seasons..
Summer
Autumn
Winter---we also got rid of the leaves
Spring