Mathematica Programming Challenge 2001
Download the Mathematica notebook of
solutions.
The Challenge: Aggregation!
Simulate a simple model of aggregation on a 2D grid: start with a single
black cell, and at each step fill in a random white neighbor of a randomly
selected existing black cell.
The Details
This is a simple model of growth sometimes called the Eden model
(specifically, a type B Eden model). The model has been used since the
1960s as a model for such things as tumor growth and growth of cities.
Here's how it works. Have a 2D grid of black and white cells (1s and 0s).
Start with a single black cell. At each step, select at random an existing
black cell that has at least one white neighbor. Then select a
neighboring white cell at random, and make it black. (By neighbor we mean
a cell above, below, left or right--not diagonal.) Your function should be
Aggregate[state, t]; state should be an array of 0s and 1s, t
should be
the number of steps, and the result should be another array of 0s and 1s,
with exactly t new black cells. Your function should allow the cluster of
black cells to get as big as it wants (i.e., the array may grow). Unfilled
cells should always be assumed white. See if you can generate a really big
cluster. Make a picture of it, using ListDensityPlot. Is it a
circle?
Have Fun!
A selection of the most efficient or most elegant solutions will be
presented at the dinner Friday night.
If You are Feeling Adventurous...
...you could try changing the criterion for growth, and seeing what shapes
you can get. For example, allow growth only when the neighbors around a
cell have some particular form. What shapes can you make?
Download this information as a notebook.
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