Transcript Slide 1

--and what can you do with it in class?
David Chan
TCM 2004
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
Fractal Dimension
Can we construct one?
Hint: Because Fractals have a self-similarity
Property, we can use boxes to measure their
Dimension.
Hint(2): Look at a ratio of number of boxes
to the size of the boxes.
Hint(last): Look at the ratio of some function
of 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.)
Box dimension is calculated using:
log( N (d , F ))
dim Box F  lim
d 0
 log(d )
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
FFF(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 F ( x) , is iterated
starting with some value, say x0 , then
an orbit is created. This orbit, or
sequence, is written as
x0 , x1  F ( x0 ), x2  F ( x0 ),
2
, xn  F ( x0 ),
n
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
An Example of systems that give attractors:
Chaos Game
http://www.shodor.org/interactive/activities/chaosgame
Examples of keeping track of
attractors
Julia Sets
Mandelbrot Sets
-Everyone’s favorite curved function:
fc ( x)  x  c
2
-Complex Plane
fc ( z)  z  c
2
-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
F ( z )  F ( z0 )
F '( z )  lim
.
z  z0
z  z0
F ( x  h  iy)  F ( x  iy)
F '( z)  lim
.
h0
h
F ( x  i( y  h))  F ( x  iy)
F '( z)  lim
.
h0
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.