1.choose your favorite positive integer a between 1 and 100 2.if a = 1, then stop 3.if a is even,...

12

1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.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. The Mathematical Kevin Bacon Game

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Page 1: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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

Page 2: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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

Page 3: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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

Page 4: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

?

Page 5: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

What is a feedback system?

Xnfunction Xn+1

Page 6: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

Can Fractals Really Arise Naturally?

Page 7: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

Fractals – The Geometry of Feedback Systems

Page 8: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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

Page 9: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

)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

Page 10: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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!

Page 11: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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

Page 12: 1.Choose your favorite positive integer A between 1 and 100 2.If A = 1, then STOP 3.If A is even, replace A by A/2 and go to step 2 4.If A is odd, replace

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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