They are completely self-similar.

They contain spirals and bifurcations.

They have periodicities (repeating patterns).

They can be based on the equation: z = z +

c

.

Each set is calculated for a single value of

c

.

The starting value of

z

is varied.

Complex numbers are used for the sets.

The starting value of

z

is plotted on the axes

x-axis - real part of

z

.

y-axis - imaginary part of

z

.

Values of

z

where the result does not go to infinity are in the Julia set and are plotted black.

The other values of

z

are plotted in color.

Provide examples of self-similarity.

Show infinite complexity.

Constructed by repeating a simple process.

Show branching and spiral patterns.

They are found in many things in nature:

Vegetables/Fruits

Plants

Bodies

Sea creatures

Environment

Space

They can have different patterns:

Branching - Bifurcation

Fibonacci Spiral

What are fractals?

How big and small fractals are -

You can zoom in on mathematical fractals for an infinite number of levels.

Computers can zoom in 16 orders of magnitude easily today.

The diameter of an atom is 10 meters.

The diameter of the solar system is 10 meters.

The difference between the two is 22 orders of magnitude.

Algebraic fractals -

What fractals are used for -

We can see fractals used in many areas of our everyday lives, such as:

Medicine

Science/Industry

Military

City design

Animation

Design movie landscapes

Design animal skins (dinosaurs)

Discovered by Benoit B. Mandelbrot in 1980.

It is not completely self-similar.

It contains spirals and bifurcations.

It has periodicities (repeating patterns).

It is based on the equation:

z = z + c

.

The equation is calculated for different values of

c

.

The starting value of

z

is always zero.

Complex numbers are used for the set.

c

is plotted on the axes

x-axis - real part of

c

y-axis - imaginary part of

c

**THE MYSTERY**

of

FRACTALS!

of

FRACTALS!

Mandelbrot Set

Zoomed in to 176 orders of magnitude.

This shows more complexity than the pictures zoomed in only a few orders of magnitude.

Julia Sets

Fractals are

geometric shapes

.

You have heard of many geometric shapes: squares, triangles and circles.

New area of math

They have no simple definition

Still being defined today

They are

infinitely complex

Created by

repeating a simple process

Special

self-similarity

property

Chaos theory -

Science discovered by Edward Lorenz in 1961 while studying weather.

Chaos is said to deal with the science of surprises.

Very small differences in the input to the equations can cause wildly different results.

Chaos deals with nonlinear equations like those used in the Mandelbrot set:

z = z + c

Fractal Fun Facts -

Albrecht Durer might have constructed the first fractal in 1525 in his book "Instruction in Measurement".

A spiral galaxy is the largest natural fractal.

Fractal Activity -

Construct a Sierpinski triangle with the help of the entire class.

Everyone builds their own Sierpinski triangle.

Then we combine all of the individual triangles into one large classroom-size Sierpinski triangle.

References -

Where fractals are found -

Fractals are found in

:

Nature (living/non-living things)

Algebra

Geometry

Medicine

Engineering

Fractal patterns

Branching (bifurcations)

Spirals

Fractals found in nature -

**By Cian and Nolan**

Geometric fractals -

Euclid of Alexandria

325-265 BC

Defined geometry in "The Elements"

(a book about geometry)

Self-Similarity

It means that no matter how far you zoom in. You always see the same shapes and patterns but smaller.

You can shrink the original shape and put it over a portion of the original shape.

Pentaflake

Spiral Galazy

Vegetables

Broccoli - Branching fractal

Romanescu - Spiral fractal

Fruits

Pineapple - Spiral fractal

Aloe - Spiral fractal

Fiddlehead Fern - Spiral fractal

Plants

Bodies

Lungs - Branching fractal

Neurons - Branching fractal

Sea Creatures

Nautilus - Spiral fractal

Sea Urchin - Spiral fractal

Enviroment

Lightning- Branching fractal

Hurricane - Spiral fractal

Rivers/Fjords -

Branching fractals

Space

Milky Way Galaxy - Spiral fractal

Sierpinski

Triangle/Carpet

Pentaflake

Koch Snowflake

Cantor Dust

Koch Curve

Sierpinski

Triangle Construction

3-D

Sierpinski Tetrahedron

(Tetrix)

3-D

Sierpinski Cube

(Menger Sponge)

Made by repeating a simple equation many times to find out if the result goes to infinity or stays finite.

They are infinitely complex.

Computers are used to help with the calculations.

Examples:

Mandelbrot Set

Julia Set

2

Mandelbrot Set

Complex Numbers

They have real and imaginary parts

An imaginary number is :

A complex number looks like:

Mandelbrot Set

Axes

Constructing the Mandelbrot Set

Julia Sets

c = -0.89 + 0.00

i

c = -0.91 + 0.31

i

c = -1.63 - 0.03

i

c = 0.03 + 0.63

i

c = 0.26 + 0.00

i

c = 0.37 + 1.08

i

c = 0.00 + 0.00

i

Medicine

Detect emphysema in lungs by using fractal patterns.

Detect cancerous tumors looking at blood vessel patterns

Science/Industry

Computer chip cooling cooling circuits.

Non-turbulent mixers used in fields such as high-precision epoxies

Military

Fractal camouflage clothing

City Design

To study the best city design to limit natural resource usage.

Rome, Italy

Suspension bridge

Cable

Cable

Construction

and

Cross-section

Cables for bridges.

Fractal antennas in cell phones and Wi-Fi systems.

Sierpinski antenna design

Cell phone with Sierpinski carpet antenna

-10

12

Mandelbrot Set

zoomed 275 orders of magnitude

Self-similarity explained

2

Chaos

Principles of chaos:

Butterfly Effect

- Input values greatly effect results

Unpredictability

- Result can't be predicted

Order/Disorder

- Study transition between order and disorder

Mixing

- Turbulance, mixed systems can't be unmixed

Feedback

- Action generates response that modifies the action

Fractals

- Infinitely complex never ending patterns

Chaos used to model systems:

In biology, economics, planetary motion

Such as weather, turbulance, fluid dynamics, traffic on roads

Chaos Explained

Chaos

Evolution of 3 starting values

Self-similarity in Geometric Fractals

Mandelbrot Set

Values of

c

that give finite

z

2

Julia and Mandelbrot Sets

Julia Sets are connected or disconnected.

When the point

z = 0

does not go to infinity (bounded) the Julia Set is connected.

When the point

z = 0

does go to infinity (unbounded) the Julia Set is disconnected.

Disconnected Julia Sets are also called Cantor Dust

Julia and Mandelbrot Sets are related for a single equation.

Values of

c

in the Mandelbrot Set give connected Julia Sets.