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1. Choose your favorite positive integer A between 1 and 100
2. If A = 1, then STOP3. If A is even, replace A by A/2 and go to step 24. If A is odd, replace A by 3A + 1 and go to step
2Count How Many Steps it takes. Your goal is to
find the A that gives you the biggest number of steps.
The Mathematical Kevin Bacon Game
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Fractals (Part 2):The Geometry of
Feedback
In which I speculate about a strange alternative-history
for mathematicsBut an initially rosy picture turns dark as the terrible
clouds of Chaos loom on the horizon.
Chaos Warrior
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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
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?
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What is a feedback system?
Xnfunction Xn+1
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Can Fractals Really Arise Naturally?
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Fractals – The Geometry of Feedback Systems
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An Example Feedback System
t
tt
n
nn 1
Growth Rate
)1( tn
Should be proportional to this
The environment can only support so many ninjas! Especially due to rampant destruction
of natural ninja habitats.
n Number of ninjas
Max ninjas the environment can support
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)1(1 tttt nrnnn
01.,3 onr
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 970
0.2
0.4
0.6
0.8
1
1.2
1.4
Ninja Population
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16111621263136414651566166717681869196
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ninja Population
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 960
0.2
0.4
0.6
0.8
1
1.2
1.4
Ninja Population (similar)1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
0
0.2
0.4
0.6
0.8
1
"Error"
MWHAHAHAHAHA!
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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
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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?