Governor’s School for the Sciences

MathematicsMathematicsDay 10

MOTD: Felix Hausdorff

• 1868 to 1942 (Germany)

• Worked in Topology and Set Theory

• Proved that aleph(n+1) = 2aleph(n)

• Created Hausdorff dimension and term ‘metric space’

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 dd with 1 < d < 2

Fractal Area

• Given a figure F and a dimension dd, what is the dd-dim’l area of F ?

• Cover the figure with a minimal number (N) of circles of radius e

• Approx. dd-dim’l area is A,d(F) = N.C(d)d

where C(d) is a constant (C(1)=2, C(2)=)

• dd-dim’l area of F : Ad(F) = lim->0 A,d(F)

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

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 don’t want A=0, we need 3(1/2)d = 1 or d = log 3/log 2 = 1.585

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)/2M+1

• Area: C(d*)3M (sqrt(2)/2M+1)d*

= C(d*) (sqrt(2)/2)d* = C(d*) 0.5773

Twin Christmas Tree

Sierpinski Carpet

3-fold Dragon

d* = log(3)/log(2)

d* = 2

d* = log(8)/log(3)

Koch Curve

d* = log(4)/log(3)

MRCM revisited

• Recall: Mathematically, a MRCM is a set of transformations {Ti:i=1,..,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

Example IFS (Koch)

1. Start with any point on the unit segment2. Randomly apply a transformation3. Repeat

Fern

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

Fern (adjusted p’s)

Lab

• Use your transformations in a MRCM and an IFS

• Experiment with other transformations

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