| Fractals appear in the world both as objects and as
time records of processes. |
| Practically every example observed involves
what appears to be some element of randomness, perhaps due to the interactions
of very many small parts of the process. |
| Think of the complicated interaction
of hydrodynamical and geological forces scuplting a fractal coastline, or of the
thousands of economic agents driving the stock market. |
| Here we investrigate
random fractals, look at various statistical properties they possess, note
how they appear in a huge variety of settings (from bacteria colonies to
clusters of galaxies), and investigate a new family of very simple random
cartoons that can be fine-tuned to exhibit a variety of statistical properties. |
| Diffusion-Limited Aggregation (DLA)
is a growth model in which cohesive particles follow Brownian motion paths (the
"diffusion" of the name) through a medium until they encounter one another and stick.
The clusters exhibit hierarchical branching, but the statistics of large clusters is
subtle and not yet completely understood. |
| What do coffee and the distribution of galaxies have in common?
Percolation is a
model consisting of cells in a lattice that randomly
take one of two states, say red and
blue. If the probability of blue is high enough, blue
clusters span the lattice. That is, blue flows (percolates) through the lattice. |
| Bacterial
Growth in Stressed Environments
Growing bacteria on media depleted of nutrient often yields colonies resembling DLA clusters.
The colony growth is limited by the nutrient diffusing in from the edge of the media, so
this is no surprise. But some propose another model: the colony shape is the result of
communication among the bacteria. |
| Large-Scale Distribution of Galaxies
Over some range of distances, the distribution of galaxies in the
universe is lumpy and clumpy: clusters, superclusters, strings, walls, huge voids. Does the
entire universe exhibit this fractal structure? Is it a result of the precise conditions of
the Big Bang, or a natural consequence of the expansion that followed? |
| Lungs The lungs are remarkable. A volume of
5 or 6 liters contains a surface area of 130 square meters, achieved through repeated branching. Evolution
discovered fractal design principles. |
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| Homework 6 |
| Homework 5 Solutions |