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Transcript of Fractals
What are some real life fractal-like structures?
Economic Patterns and Modeling
by Una Boyle,
and Genevieve Fowler
What is a fractal, anyway?
A fractal is
Iterated Function Systems (IFS)
This is an easy method for creating simple fractals.
You begin with an initial
(like a triangle) and then create a set of "
" for it to replicate itself into a pattern.
Can't get enough
shape is made of smaller copies of itself. The copies are similar to the whole: the same shape but different size.
Each piece of the fractal is a smaller version of the whole thing!
Is a line a fractal?
6. Is this
Is it self similar? Should that count?
What do we mean by rules?
Basically, we have several parameters (
r, s, θ, φ, e, and f
) that we can manipulate and each has a different effect on how the shape replicates itself. To create a rule, we
assign a value
to each of these parameters.
Review: In school, you deal with
dimensions. Lines are 1 dimensional. Circles and parabolas you draw in the Cartesian coordinate system (with x and y axes) are 2 dimensional. Cubes and cones are 3 dimensional.
don't need to have integer dimensions.
The Moran Equation
First, let's redefine a fractal formally.
That's right. Not only can fractals have non-integer dimensions, they can also have more than one dimension.
Make your own!
is a word used to describe something that has the same pattern averaged over
Thanks for coming
to our class!
1. Start at the center dot.
2. Pick the next random number on your slip.
3. Put down a dot halfway between your current dot and the corner corresponding to the random number you picked.
4. Choose a new random number
You can contact us at:
The Julia Set
The Mandelbrot Set
A shape is
if it can be broken into
pieces, overlapping at most along edges, scaled by a factor of
If every piece is the same dimension as the original:
d = Log(N)/Log(1/r)
If pieces are of varying dimensions:
So... dimension is just an exponent relating
?! And it's not even always an integer?!
Yep! And if you think that's crazy, wait til you hear what's next...
The Sierpinski Triangle
The Koch Snowflake
The Cantor Set
Yale Professor Michael Frame is one of the world's experts on fractals
Visit his website (http://classes.yale.edu/fractals/)
Or watch his awesome
talk on YouTube
Fractals are absurdly useful in lots of different fields of research...lots of new ideas are still being created
Getting more specific: What
as a fractal?
1. No physical fractal is infinitely self similar, but to count something as a fractal in the real world, it must exhibit at least
a few levels of self-similarity.
2. There must be
at least two self-similar copies
of the original image each time we magnify to another level of the fractal.
Just a repeating pattern does not mean it is a fractal.
The Koch Curve
Now you try
It turns out all fractals have an
Related: Coastlines, lung air pathways
However, fractals have