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

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.