David Chan

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--and what can you do with it in class?
David Chan TCM 2004
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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

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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.

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Examples
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Fractal Dimension

• Can we construct one?

Hint Because Fractals have a self-similarity Prop
erty, 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
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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

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Dimension (cont.)
Box dimension is calculated using
where N(d,F) is the smallest number of sets of
diameter d which can cover F.
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How are fractals constructed?

• Geometrical Process

• Function Composition

• Function Attractors

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Koch Snowflake
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Sierpinskis Triangle
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Cantors Middle Thirds Set

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L-systems
Example

• Start off with a rule

F?FF(LF)(RF)

• And an initial string

F

• Then compose/iterate

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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)(L
F)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(R
F)(LF)(RFF(RF)(LF))(LFF(RF)(LF)))(RFF(RF) (LF)FF(R
F)(LF)(RFF(RF)(LF))(LFF(RF)(LF))FF(RF)(LF)FF(RF)(L
F)(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(R
F)(LF))(LFF(RF)(LF))))
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Attractors

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

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• 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

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An Example of systems that give attractors
Chaos Game
http//www.shodor.org/interactive/activities/chaos
game
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Examples of keeping track of attractors

• Julia Sets

Mandelbrot Sets
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-Everyones favorite curved function
-Complex Plane
-Complex Arithmetic
-Graphing Complex Functions
-Complex DERIVATIVES!
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COMPLEX DERIVATIVES!

• Definition For a complex function F(z), we
  define its complex derivative, F(z), to be

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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.