R. Robinson 4/9/01
After a protien molecule is manuactured in a cell (by transcribing DNA sequences to amino acid sequences) it folds into a 3-dimensional shape in a way that basically minimizes its entropy. The protein folding problem searches for an efficient algorithm that describes this folding process. See http://grserv.med.jhmi.edu/~przytyck/
The brachistocrone problem searches for the decreasing curve between two points such that a car will roll down it the fastest. The answer is known to be the cycloid (not a straight line). Using a piecewise linear curve, find a numerical approximation of the cycloid http://grserv.med.jhmi.edu/~przytyck/
Represent a knot as a piecwise linear closed curve. Use a multidimensional optimization technique to find small total curvature or small total energy configurations.
Fractals can be used as a method for image compression. See if you can implement this.
Write an algorithm to compute the fractal dimension of a subset of the plane. Apply this to some fractals (eg, I can tell you about some fractals that arise in the theory of camplex beta transformations).
Use the DFT and DWT to process two dimensional images.
Look into data encryption and implement an encryption scheme.
Look into a data compression algorithm and implement a data compression method.
In his book The Evolution of Cooperation, political scientist Robert Axelrod studied iterated prisoner's dilemma games. See also some of the paopers by Karl Sigmund and Josef Hofbauer.
In the book The Algorithmic Beauty of Sea Shells (Virtual Laboratory) by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler, models for the coloration of sea shells are discussed. Similar models using cellular automata have been described by Steve Wolfram (the primary author of Mathematica).
The Jacobi-Perron algorithm is an example of a multidimensional continued fraction algorithm. It has been proven to have an invariant density, but an explicit formula for it has never been found. A closely related algorithm, called the modified Jacobi-Perron algorithm has a well known invariant density (i.e., the formula is known). There are many related algorithms, some of which have known densities and some of which don't. Computer experiments provide a good way to see what invariant densities look like and to test conjectures about them.
One of the most famous stochastic models is the Ising model in statistical physics. It can be solved numerically using the famous Metropolis algorithm.
Look into algorithms for constructing Penrose like tilings of the plane, either by substitution (inflation) or by the "projection method".