Careful experiments would involve making notes of the settings that produce each pattern you observe. Then you could test reproducibility - will the same pattern appear if you use the same settings, and sensitvity to initial conditions with the same settings, change the ambient light level in the room. Does this give rise to a different pattern?
Driven IFS and data analysis
One way to observe correlations in a noisy data sequence is to make a "driven IFS" picture. Construct from the data sequence a sequence of 1s, 2s, 3s, and 4s by partitioning the raw data into four equal size bins, where bin 1 includes the set of data with the smallest values, bin 2 the next smallest, and so forth. Define four transformations by
T1(x,y) = (x/2, y/2),
T2(x,y) = (x/2 + 1/2, y/2),
T3(x,y) = (x/2, y/2 + 1/2), and
T4(x,y) = (x/2 + 1/2, y/2 + 1/2).
Take as a starting point x = 0.5, y = 0.5. Generate a picture by applying the transformations that correspond to the sequence of binned data. In other words, if the first binned data value is 3, apply T3 to (0.5, 0.5) -- yielding (0.25, 0.75); if the next binned data value is 2, apply T2$ to (0.25, 0.75); and so on. You might apply driven IFS to look for dynamical exclusions in financial data, daily high temperatures over a year, or whatever you like.
IFS and nature
Take pictures of natural objects -- trees, clouds, ferns, flowers -- and experiment with finding IFS rules to produce convincing forgeries. Comment on how changing the parameters affects the pictures. Do you observe any patterns.
Fractals in poetry
Try to apply Pollard-Gott's analysis of Wallace Stevens to other poets, perhaps using repeated ideas or sounds instead of root words. Another possibility is to find a sensible way to analyze poetry by driven IFS.
Fractals and chaos in literature
Discuss the occurrence of fractals or chaos -- direct or as metaphor -- in literature. Find your own sources, or consider Vladimir Nabokov's The Eye or ``Ultima Thule,'' Jorge Luis Borges' ``The Garden of Forking Paths,'' Gloria Naylor's Mama Day, Flan O'Brien's The Third Policeman, Kate Wilhelm's Death Qualified, John Updike's Roger's Version, or Michael Crichton's Jurassic Park.
Analyze the occurrence of fractals in paintings: Max Ernst, Oscar Dominguez, or others. Why do fractals occur in these paintings? Another possibility is to produce and analyze fractal paintings of your own. Variants of the goo experiments, or the flow of one ink into another, are possibilities.
Fractals and chaos in financial data
Obtain several years' time series for your favorite financial data. Then tune the fractal time model to match your time series. Do the model properties suggest anything about the source of the data.
Cellular automata experiments
Explore self-organization or sensitivity to initial conditions in cellular automata.
Chaos for different functions
Chaos is exhibited by many functions in addition to the tent and logistic maps. But the structure of the relations between chaos and order can be radically different from these cases. Investigate the structure of the bifurcation diagram for a function such as s*x2*(1 - x), s*x*(1 - x2), s*x*(1 - x)2, etc.
Mandelbrot sets for different functions
Instead of z2 + c, how does the Mandelbrot set for z3 + c, or z4 + c, or z3 - 3*z + c look? How does this shape compare with the standard Mandelbrot set? What are the main differences and smilarities you see?
Measuring dimensions of natural objects
Use boxcounting to measure the dimensions of natural objects. (Use transparencies with grids to make the counting easier.) Some possibilities include cloud perimeters, coastlines, river meanders (from photograohs and topographic maps), and cracks in concrete (do a rubbing of the crack and measure that). Interpret your results.
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