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Well, it's that time of year... I need a science project. This year, I'd like to do something in Computer Science. Could be CA, could be genetic algorithms, maybe even neural networks if I tried really hard. What I want to figure out is a project with an independent variable (IV) and a dependent variable (DV). The independent variable is changed to see it's effect on the dependent variable. Project titles are 'The Effect of [IV] on [DV]' So, if I made a project called 'The Effect of Soup Density on the Number of Uncommon Still Lifes', I would make many soups of different densities and find how many uncommon still lifes each soup density makes per 100 square pixels.

So... Any ideas? As usual I am baffled and can't think of anything.

The effect of the parameters on reaction-diffusion systems?

http://ferkeltongs.livejournal.com/32025.html

Reaction-diffusion systems were invented by Turing to explain biological processes of morphogenesis and pattern formation. They are similar to cellular automata, in that they operate on a Euclidean plane, and can be simulated to a reasonable accuracy on a Cartesian grid.

Also, the patterns produced are aesthetically pleasing -- more than can be said about cellular automata!





This is a good tool to experiment with:

http://texturegarden.com/java/rd/



The six parameters on the right allow you to alter basic variables in the system. You can then investigate which configurations of variables result in different emergent patterns, such as zebra stripes, or amoeboid fission.


And there's a lot of mathematics behind the scenes you can mention in your project. It is based on partial differential equations (have you encountered them yet?), which can be solved approximately using Euler's method, or one of its variants.


As well as biology, there are also connections to chemistry:

http://en.wikipedia.org/wiki/B-Z_reaction

You could prepare a conical flask and/or Petri dish of B-Z juice and include it as part of your project.

Thanks! I've seen Reaction-Diffusion before but never considered it as a science project option until now. Could you explain where it's found in nature exactly? And I'll look up partial differential equations...

Could you explain where it's found in nature exactly?

It's a natural chemical process, and takes place in many reactions, the most interesting being the B-Z reaction. (The colour continually fluctuates, and it resembles a cyclic cellular automaton.)

Alan Turing originally developed the model to explain how patterns form on animals and plants, based on chemicals reacting and diffusing between cells, hence the name: reaction-diffusion system. The model based on partial differential equations is the idealised limit, where the size of the cells approach zero.

And I'll look up partial differential equations...

The formal definition of a PDE is an equation in the partial derivatives of a variable with respect to other variables. The canonical PDE is the beautiful wave equation:

d²U/dx² + d²U/dy² = d²U/dt²

(U represents the height of the wave at the location xi + yj, where i and j are the unit vectors, directed east and north respectively.)

That's a second-order partial differential equation. Second order differential equations are usually reversible, similar to certain cellular automata.


The equation for heat transfer in one dimension is, if I remember correctly:

dT/dt = d²T/dx²

(T represents the temperature at the point x.)

That encompasses both first-order and second-order components. In fact, it is the simplest such PDE.


Informally, a partial differential equation is an analogue of cellular automata, where space, time and states all become continuous, rather than discrete. One could, theoretically, design a PDE version of Conway's Game of Life, and use Euler's method to simulate it.

Another thing that you could do is a study of glider synthesis in relation to the product vs. the position and number of gliders.