Experimental Math

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Experimental Math

• Cellular Automata

Rikke Bendlin Mia Grifford Geetika Gupta Klodiana
Hajdari Sarah Loos
Amy McNabb Raquel Phillips Crystal
Reidinger Elizabeth Thomas Henry Cohn
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Overview

• Cellular Automata
• Entropy
• Integrating the two

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Cellular Automata

• A discrete model studied in computability theory,
  mathematics, and theoretical biology.
• It consists of a regular grid of cells, each in
  one of a finite number of states

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Cellular AutomataUpdate Rule
Rule 30
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Entropy

• A measure of disorder or randomness in a closed
  system
• Pi is the probability of case i.
•  
• ? Pi log2 (1/Pi)

n
i 0
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The Goal

• Calculate the entropy
• See how it relates to visual patterns based on
  the rules

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Entropy Window Sizes
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Rule 96

• Ent1 is 0.0?
• Ent2 is 0.0?
• Ent3 is 0.0?
• Ent4 is 0.0

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Rule 99

• Ent1 is 1.00000?
• Ent2 is 0.52036?
• Ent3 is 0.36046?
• Ent4 is 0.27041

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Rule 106

• Ent1 is 1.00000
• Ent2 is 1.00000
• Ent3 is 1.00000
• Ent4 is 0.87515

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Rule 110

• Ent1 is 0.98540?
• Ent2 is 0.97914?
• Ent3 is 0.96844?
• Ent4 is 0.76504

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Rule 115

• Ent1 is 0.95591?
• Ent2 is 0.95510?
• Ent3 is 0.86780?
• Ent4 is 0.56536

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Rule 129

• Ent1 is 1.00000?
• Ent2 is 0.90679?
• Ent3 is 0.83926?
• Ent4 is 0.78901

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Rule 130

• Ent1 is 0.63670?
• Ent2 is 0.61427?
• Ent3 is 0.58823?
• Ent4 is 0.44118

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Future Work

• Bigger window sizes
• More states
• Different rules
• More classification

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Real World Applications

• CA can be used in algorithmic music compositions.
  Many musicians have used it for their
  compositions
• Many games are built on the principle of CA like
  Tetris
• And you can build pretty pictures if thats all
  you want D

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Questions or Comments?