choose your favorite positive integer a between 1 and 100 if a = 1, then stop
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
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
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
?
What is a feedback system?
Xnfunction Xn+1
Can Fractals Really Arise Naturally?
Fractals – The Geometry of Feedback Systems
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
)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
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!
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
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?