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Simple case [of three-body problem]
The position of the idealized planet in the case shown satisfies the differential equation
δ tt z[t] -z[t]/(z[t] 2 + (1/2 (1 + e Sin[2 π t] ) 2 ) 3/2
where e is the eccentricity of the elliptical orbit of the stars ( e = 0.1 in the picture).
And typically one gives this rate as a simple formula that depends on the gray level at each point in space, and on the rate at which that gray level changes with position.
In the particular case shown, the rules are simply set up to shift every color one position to the left at each step.
And what the pictures demonstrate is that even if the initial position of this mass is changed by just one part in a hundred million, then within 50 revolutions of the large masses the trajectory of the small mass will end up being almost completely different.
If one looks at individual particles, then changing the position of even one particle will typically have an effect that spreads rapidly.
And as an example of a simple approach to modelling this, one can consider having a collection of discrete eddies that occur at discrete positions in the fluid, and interact through simple cellular automaton rules.
But in 1981 it so happened that I had for some years been deeply involved in both practical computing and basic science, and I was therefore in an almost unique position to apply ideas derived from practical computing to basic science.
And indeed I believe that it is only with the discoveries in this book that one is finally now in a position to develop a real understanding of what randomness is.
And this means that if black is represented by 1 and white by 0, one can then give an explicit formula for the color of the square at position x on row y : it is simply (1 - (-1)^Binomial[y, x])/2 .
Picture (b) then shows another version of this same evolution, but now rearranged so that each element stays in the same position, rather than always shifting to the left at each step.