Section 6.1Images

Viewing a Gallery of Fractals.

Look for patterns.

Question of the Day

How can you convince your parents that you’re eating enough broccoli?

Real or fake?

The Sierpinski Triangle

How many Quackers do you see?

The Fern

Julia Sets

Section 6.2The Infinitely Detailed Beauty of Fractals

How to create works of infinite intricacy through repeated processes.

It’s not where we begin, it’s how we get there.

Question of the Day

What do you need to know to predict the world’s population in the future?

Self-Similarity

The characteristic of looking the same as or similar to itself under increasing magnification.

A process of repeated replacement.

Koch’s Kinky Curve

A process of repeated replacement.

Sierpinski Triangle

A process of repeated replacement.

Menger Sponge

A process of repeated replacement.

Barnsley’s Fern

More on the Sierpinski Triangle!

The final bow…

The Chaos Game

Whatever number you generate, move halfway from where you are toward that numbered vertex and make a dot. Then roll again to determine the next number… and repeat the process.

Section 6.3Between Dimensions

Can the Dimensions of Fractals Fall through the Cracks?

Start with the simple and familiar.

Look for patterns.

Apply patterns to new settings.

Question of the Day

How do you turn a triangle into a snowflake?

What is a dimension?

1-dimensional: A line segment

2-dimensional: A filled in square

3-dimensional: A solid cube

4-dimensional: ?

In search of a pattern…

Original Object

Dimension of the object

Scaling Factor to Make a Larger Copy

Number of Copies Needed to Build the Larger Copy

Line

Square

Cube

Section 6.4The Mysterious Art of Imaginary Fractals

Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane

Don’t be afraid to look and think

before you try to understand.

Question of the Day

What’s the story behind this picture?

Julia Sets

Mandelbrot Sets

A process in the plane.

Review of Complex Numbers.

2

3

4

123

1

.

.

.

i

i

i

i

i

Review of Complex Numbers.

2

a bi c di

a bi c di

a bi

Visualizing Complex Numbers

What is a Mandelbrot Set?

An object that captures information about the collection of all (a + bi) – Julia Sets.

Section 6.5The Dynamics of ChangeCan Change Be Modeled by

Repeated Applications of Simple Professes?

The best predictors of where you will be are where you are now and which way you’re going.

Question of the Day

How close is your calculator’s answer to the correct answer?

Repeat, Repeat, Repeat

• Start somewhere.

• Apply a process and get a result.

• Apply the same process to that result to get a new result.

• Apply the process again to the new result to get a newer result.

• Repeat patiently and persistently, forever.

Conway’s Game of Life

1. A living square will remain alive in the next generation if exactly two or three of the adjoining eight squares are alive in this generation; otherwise, it will die.

2. A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.

Play the Game of Life!

Patterns in types of Populations

• Explosion

• Extinction

• Stable

• Periodic

• Migratory

Population Density

Population Density =

The change in population density is the change from year n to n+1:

the number at year

the maximum sustainabable population n

nP

nP

1n nP P

The Rate of Change of Population

1n n

n

P P

P

Section 6.6Predetermined Chaos

How Repeated Simple Processes Result in Utter Chaos

Keep alert for significance even in ridiculous ideas.

Question of the Day

What is the dimension of a cloud?

Experiment1. Type in a random number.

2. Multiply it by 180.

3. Hit the SIN key and write down your answer in the second column next two 2. Keep all decimal places.

4. Repeat steps 2 and 3 for each new number you generate. Continue to record your results until you have 25 numbers in the second column.

A variation on the experiment1. Enter the exact same first decimal

number as before and repeat the process five times.

2. Clear your calculator of the previous result.

3. For the sixth number use the fifth number rounded to six decimals places.

4. Now, repeat the process as before until you have 25 numbers in the second column.

Predicting Future Populations

The Verhulst Model

• = population in year n

• = population for the next year.

1 3 1n n n nP P P P

nP

1nP