Governor’s School for the Sciences Mathematics Day 10

MOTD: Felix Hausdorff 1868 to 1942 (Germany) Worked in Topology and Set Theory Proved that aleph(n+1) = 2 aleph(n) Created Hausdorff dimension and term ‘metric space’

Fractal Dimension dA 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 < d < 2

Fractal Area d dGiven a figure F and a dimension d, what is the d-dim’l area of F ? Cover the figure with a minimal number (N) of circles of radius e dApprox. d-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 : A d (F) = lim  ->0 A ,d (F)

Fractal Area (cont.) If d is too small then A d (F) is infinite, if d is too large then A d (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 fractal Then since each part is the image of the whole under the transformation: A = 3(1/2) d A Since 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 3 M squares of size (1/2) M x(1/2) M covered by 3 M circles of radius sqrt(2)/2 M+1 Area: C(d*)3 M (sqrt(2)/2 M+1 ) d* = C(d*) (sqrt(2)/2) d* = C(d*)

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 {T i :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 segment 2. Randomly apply a transformation 3. 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: 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