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.
The first fractals were discovered by a french Mathematician
named Gaston Julia who discovered them decades before the
advent of computer graphics.
Julia’s work was
rediscovered by
Benoit
Mandelbrot.
The most famous
of all fractals is the
Mandelbrot set.
...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
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?
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)
Aristid Lindenmeyer
Aristid Lindenmeyer invented LSystems to model plant growth.
See the “fractal” he is holding?
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.
This is the crudest version. “Nature made of
triangles.”
In this picture, the grid of triangles is a bit finer
The triangles are finer still and the mountains in the distance
begin to look a bit better.
The next step. The mountains in the background
look like real mountains.
Now we add some texture to the triangles.
Here we add some crude clouds and improve the landscape.
Here we add some crude trees, and improve the
landscape and the clouds.
This is the best resolution I got.
Summer
Autumn
We can
make some
artistic
choices of
color to
give us the
same
landscape
in different
seasons..
Winter---we also got
rid of the leaves
Spring