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…
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.