AN INVESTIGATION INTO THE NATURE OF GROWTH IN ELEMENTARY CELLULAR AUTOMATA To Live and Die in CA by Jane Curnutt Committee: Yasha Karant, advisor Ernesto Gomez Keith Schubert

INTRODUCTION TO CA (CELLULAR AUTOMATA) To Live and Die in CA

CA Vocabulary Deterministic Synchronous Totalistic Grid (like a piece of graph paper) Neighborhood  Moore Neighborhood  von Neumann Neighborhood Calculation Radius (Radius) Target Square Rules: birth=n1 death>n2

To Live and Die in CA Neighborhoods Moore von Neumann

To Live and Die in CA radius=3radius=1 target square

To Live and Die in CA Neighborhoods Radius = 1Radius = 3

BIOLOGICAL CA MODELS To Live and Die in CA

Cellular Automata – Why do we want to do this? 5 steps 40 steps

Biovermiculation with Discontinuities CA Simulation - Penny Boston To Live and Die in CA

CA Model of Bioverm Cave Bioverm photo courtesy of Penny Boston

THE NATURE OF GROWTH IN CELLULAR AUTOMATA To Live and Die in CA

Birth=1 Radius=1 Death=0 Time=4

To Live and Die in CA

Birth=1 Radius=1 Death=0 Time=5

To Live and Die in CA

Birth=1 Radius=1 Death=0 Time=4

To Live and Die in CA

Birth=1 Radius=1 Death=0 Time=4

To Live and Die in CA

Conclusions Growth only takes place at endpoints and vertices Growth is always perpendicular to the growth in previous time step Growth is recursive

FRACTAL CA To Live and Die in CA

Fractal CA Birth=1 Death>2 Radius=1 Line start

To Live and Die in CA Fractal CA Birth=1 Death>2 Radius=1 Line start This time step: Blue=new growth Red=death Green=residual growth

To Live and Die in CA Fractal CA Birth=1 Death>3 Radius=1 Line start

To Live and Die in CA This time step: Blue=new growth Red=death Green=residual growth Fractal CA Birth=1 Death>1 Radius=1 Square start

CA L-SYSTEM GRAMMAR To Live and Die in CA

L-s ystem CA Graphical Forms & Symbols

To Live and Die in CA L-system CA Symbols: Modifiers & Movement

To Live and Die in CA Why we need an SP character in the grammar? Square Inside the 2r+1 Radius red - neighborhood red - target square green - current time step blue – growth next time step

To Live and Die in CA red - neighborhood red - target square green - current time step blue – growth next time step Square Outside the 2r+1 Radius

CA L-SYSTEM PRODUCTIONS To Live and Die in CA

FUTURE RESEARCH DIRECTIONS To Live and Die in CA

 Extend CA grammar to include death rules.  Investigate the relationship between CA grammar and display grid, with different grids.  Investigate collisions between patterns, and whether CA grammar can describe them.

To Live and Die in CA Special thanks to Chris McKay, Principle Investigator for Extreme Environments, NASA, Ames and Penny Boston, NMIT Associate Director, National Cave and Karst Research Institute, for their support and encouragement.

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