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Fractals

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Rachel Lawrence

on 24 November 2013

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Transcript of Fractals

Fractals in Nature

What are some real life fractal-like structures?
Coastlines
Brains/Hearts/Lungs
Leaves/Trees/Forests
Satellite images
Economic Patterns and Modeling
...Invisibility cloaks?!
by Una Boyle,
Rachel Lawrence,
and Genevieve Fowler

Fractals
Fractals 101
Fractal Math!
Making Fractals!
What is a fractal, anyway?
A fractal is
self-similar
.
Famous Fractals
Iterated Function Systems (IFS)
This is an easy method for creating simple fractals.

You begin with an initial
shape
(like a triangle) and then create a set of "
rules
" for it to replicate itself into a pattern.


Can't get enough
Fractal Geometry?
A
self-similar
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!
DISCUSS!
Is a line a fractal?
6. Is this
presentation
a fractal?
oh snap
so meta
so amaze
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.

Software: http://classes.yale.edu/fractals/software/software.html
Fractal
Dimension
Review: In school, you deal with
integer
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.

Well, fractals
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.

Multi
-fractals?
Multiple
dimensions?!

Make your own!
Ergodicity
is a word used to describe something that has the same pattern averaged over
TIME
as over
SPACE
Thanks for coming
to our class!
THE INSTRUCTIONS
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
and repeat.
You can contact us at:
rachel.lawrence@yale.edu
una.boyle@yale.edu
genevieve.fowler@yale.edu
The Julia Set
The Mandelbrot Set
A shape is
self-similar

if it can be broken into
N
pieces, overlapping at most along edges, scaled by a factor of
r
If every piece is the same dimension as the original:
(derivation)
d = Log(N)/Log(1/r)
If pieces are of varying dimensions:
Examples!
So... dimension is just an exponent relating
scaling
and
size
?! 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
TedxYale
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
counts
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.
3.
Just a repeating pattern does not mean it is a fractal.

Source:
http://classes.yale.edu/fractals/panorama/Misc/CommonMistakes/CommonMistakes.html
2.
3.
4.
1.

5.
Perimeter
The Koch Curve

(time
series)
Now you try
It turns out all fractals have an
infinite
perimeter!
Related: Coastlines, lung air pathways

However, fractals have
finite
area.
Full transcript