Complicated behavior generated by simple computation
Stephen Wolfram gave an engaging TED talk about his work with Mathematica and Wolfram Alpha. Among other things, he pointed out how very simple calculations can lead to extremely complex behavior. (He goes on to discuss his ambitions for computing the entire physical universe. Say what you want, you should give the guy credit for his wild ideas ...)
As an example, he showed several cellular automata.
What other examples of simple calculations do you know that produce fascinating results?
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- Start at point (x, y) on the real plane.
- Repeat assignment (x, y): = (y + 1 - ax², bx), for some constants a and b.
Often used a = 1.4 and b = 0.3. For these values, the behavior is chaotic, and all the points seem to eventually converge to the following shape, called the Hénon attenuator:
This shape has fractal properties.
I say "appear" twice because none of these observations have been mathematically proven.
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- Start with any positive integer.
- If the current number is even, divide it by 2. If it is odd, multiply by 3 and add 1.
- Repeat until you reach 1.
You are supposed to reach 1 and this has been experimentally proven for large numbers (up to 5.7 * 10 ^ 18), but has not been mathematically proven.
Even for fairly small numbers, this can increase dramatically until it finally crashes to 1.
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