Exploring Fractal Dimensions and the Contributions of Felix Hausdorff

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) = 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 d with 1 < d < 2

Fractal Area • Given 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 • Approx. d-dim’l area is Ae,d(F) = N.C(d)edwhere C(d) is a constant (C(1)=2, C(2)=p) • d-dim’l area of F : Ad(F) = lime->0 Ae,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)dASince 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 3-fold Dragon d* = log(3)/log(2) d* = 2 Sierpinski Carpet Koch Curve d* = log(4)/log(3) d* = log(8)/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

Exploring Fractal Dimensions and the Contributions of Felix Hausdorff