David ChanTCM 2004

--and what can you do with it in class?

Outline

• What are Fractals?-Build a Fractal Dimension-Measure the Fractal Dimension of different

objects

• How are Fractals constructed?-Basic Fractals and their properties

-L-systems and Function Composition/Iteration-Derivatives and the Complex Plane

• Summary

What is a Fractal?

• A rough, fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole.—Benoit Mandelbrot

• (Mathematical) A set of points whose fractal

dimension exceeds its topological dimension.

• “An object whose dimension is not an integer.”

Examples

Can we construct one?

Fractal Dimension

Hint: Because Fractals have a self-similarityProperty, we can use boxes to measure theirDimension.

Hint(2): Look at a ratio of number of boxesto the size of the boxes.

Hint(last): Look at the ratio of some functionof the number of boxes to the size of the boxes

Fractal Dimension?

• Try some basic objects.

• Try some fractal objects!

• Does it make sense?

• Oh well, try again.

• Due to time constraints the answer is…

http://www.logosol.com/pictures/seidellcherrylogpile.jpg

Dimension (cont.)

0

log( ( , ))dim lim

log( )Boxd

N d FF

d

Box dimension is calculated using:

where N(d,F) is the smallest number of sets of diameter d which can cover F.

How are fractals constructed?

• Geometrical Process

• Function Composition

• Function Attractors

Koch Snowflake

Sierpinski’s Triangle

Cantor’s Middle Thirds Set

• • • •

L-systems

Example:

• Start off with a rule

FFF(LF)(RF)

• And an initial string

F

• Then compose/iterate

F

FF(RF)(LF)

FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF))

FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF)) FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF))(R FF(RF)(LF) FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF)))(L FF(RF)(LF)FF(RF)(LF)(R FF(RF)(LF))(L FF(RF)(LF))

FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))(RFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(LFF(RF)(LF)FF(RF)(LF)(RFF(RF)(LF))(LFF(RF)(LF))))

Attractors

When a function, say , is iterated starting with some value, say , then an orbit is created. This orbit, or sequence, is written as

( )F x0x

20 1 0 2 0 0, ( ), ( ), , ( ),n

nx x F x x F x x F x

Under certain conditions, orbit can converge (or limit on) a particular set of point(s). These sets are called attractors.

Types of attractors:

• Fixed points

• Periodic orbits

• Strange attractors

Chaos Game

http://www.shodor.org/interactive/activities/chaosgame

An Example of systems that give attractors:

Examples of keeping track of attractors

Julia Sets Mandelbrot Sets

-Everyone’s favorite curved function:

2( )cf x x c -Complex Plane

2( )cf z z c -Complex Arithmetic

-Graphing Complex Functions

-Complex DERIVATIVES!

COMPLEX DERIVATIVES!

Definition: For a complex function F(z), we define it’s complex derivative, F’(z), to be

0

0

0

( ) ( )'( ) lim .

z z

F z F zF z

z z

0

( ) ( )'( ) lim .

h

F x h iy F x iyF z

h

0

( ( )) ( )'( ) lim .

h

F x i y h F x iyF z

ih

Summary• Algebra/Geometry-Look at fractals and do

simple calculations. Play with the Chaos game.

• Precalculus-Shifting/Stretching pictures, L-systems and composition, and do some numerical experiments.

• Calculus-Talk about attractors and complex differentiation.• Beyond Calculus-Proofs, write programs to create fractals.