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Choose your favorite positive integer A between 1 and 100 If A = 1, then STOP

The Mathematical Kevin Bacon Game. Choose your favorite positive integer A between 1 and 100 If A = 1, then STOP If A is even, replace A by A/2 and go to step 2 If A is odd, replace A by 3A + 1 and go to step 2

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Choose your favorite positive integer A between 1 and 100 If A = 1, then STOP

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  1. The Mathematical Kevin Bacon Game • Choose your favorite positive integer A between 1 and 100 • If A = 1, then STOP • If A is even, replace A by A/2 and go to step 2 • If A is odd, replace A by 3A + 1 and go to step 2 Count How Many Steps it takes. Your goal is to find the A that gives you the biggest number of steps.

  2. Fractals (Part 2):The Geometry of Feedback In which I speculate about a strange alternative-history for mathematics But an initially rosy picture turns dark as the terrible clouds of Chaos loom on the horizon. Chaos Warrior

  3. What you should know after today • You should be able to explain what a “feedback system” is • You have a 1st idea of what “Chaos” means and how Chaos makes simulation on Computers difficult

  4. ?

  5. What is a feedback system? Xn+1 Xn function

  6. Can Fractals Really Arise Naturally?

  7. Fractals – The Geometry of Feedback Systems

  8. An Example Feedback System The environment can only support so many ninjas! Especially due to rampant destruction of natural ninja habitats. Number of ninjas Max ninjas the environment can support Growth Rate Should be proportional to this

  9. MWHAHAHAHAHA!

  10. So What’s the Deal With Chaos? • Small deviations expand, so errors multiply • Eventually the noise overwhelms the signal • Because computers can only represent numbers with limited precision, they are very vulnerable to chaos

  11. Questions • What is an example of a feedback system? • Chaos has to do with errors multiplying. Since computers can add/subtract/multiply/divide perfectly, why is there a problem with chaos on computers?

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