Governor

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Governors School for the Sciences

• Mathematics

Day 10
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MOTD Felix Hausdorff

• 1868 to 1942 (Germany)
• Worked in Topology and Set Theory
• Proved that aleph(n1) 2aleph(n)
• Created Hausdorff dimension and term metric
  space

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Fractal Dimension

• A fractal has fractional (Hausdorff) dimension,
  i.e. to measure the area and not get 0 (or length
  and not get infinity), you must measure using a
  dimension d with 1 lt d lt 2

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Fractal Area

• Given a figure F and a dimension d, what is the
  d-diml area of F ?
• Cover the figure with a minimal number (N) of
  circles of radius e
• Approx. d-diml area is Ae,d(F)
  N.C(d)edwhere C(d) is a constant (C(1)2,
  C(2)p)
• d-diml area of F Ad(F) lime-gt0 Ae,d(F)

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(No Transcript)
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Fractal Area (cont.)

• If d is too small then Ad(F) is infinite, if d
  is too large then Ad(F) 0
• There is some value d that separates the
  infinite from the 0 cases
• d is the fractal dimension of F

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Example
Let A be the area of the fractalThen since each
part is the image of the whole under the
transformation A 3(1/2)d ASince we
dont want A0, we need 3(1/2)d 1 or d
log 3/log 2 1.585
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Example (cont.)

• Unit square covered by circle of radius sqrt(2)/2
• 3 squares of size 1/2x1/2 covered by 3 circles of
  radius sqrt(2)/4
• 9 squares of size 1/4x1/4 covered by 9 circles of
  radius sqrt(2)/8
• 3M squares of size (1/2) M x(1/2)M covered by 3M
  circles of radius sqrt(2)/2M1
• Area C(d)3M (sqrt(2)/2M1)d C(d)
  (sqrt(2)/2)d C(d) 0.5773

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Twin Christmas Tree
3-fold Dragon
d log(3)/log(2)
d 2
Sierpinski Carpet
Koch Curve d log(4)/log(3)
d log(8)/log(3)
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MRCM revisited

• Recall Mathematically, a MRCM is a set of
  transformations Tii1,..,k
• This set is also an Iterated Function System or
  IFS
• Difference between MRCM and IFS is that the
  transformations are applied randomly to a
  starting point in an IFS

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Example IFS (Koch)
1. Start with any point on the unit segment2.
Randomly apply a transformation3. Repeat
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Fern
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Better IFS

• Some transformations reduce areas little, some
  lots, some to 0
• If all transformations occur with equal
  probability the big reducers will dominate the
  behavior
• If the probabilities are proportional to the
  reduction, then a more full fractal will be the
  result

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Fern (adjusted ps)
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Lab

• Use your transformations in a MRCM and an IFS
• Experiment with other transformations

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Project

• Work alone or in a team of two
• Result 15-20 minute presentation next Thursday
• PowerPoint, poster, MATLAB, or classroom activity
• Distinct from research paper
• Topic Your interest or expand on class/lab idea
• Turn in Name(s) and a brief description Thursday