Governor’s School for the Sciences Mathematics Day 14

MOTD: Srinivasa Ramanujan 1887 to 1920 (India) Worked in analytic number theory Self-educated genius

1D Results

Top 9 Scores Jennifer 442 Matt 444 Steve 453 Sam 456 Charlie F 459 Michelle 469 Stuart 484 Charlie W542 Austin 549

Prize List (So far) 4 Einstein Posters 2 Calculus Books 2 Rattle Backs Origami Book 5 Math Books (logic, writing proofs, algebra, scientific computing, literature) I Love Math T-shirt Various puzzles Various pens and pencils Stickers, bookmarks, etc.

2D Cellular Automata Associate the cells with the (infinite) latice (i,j), i,j = …,-2,-1,0,1,2,… Two types of neighborhoods: von Neumann {(k,m) : |k-i|+|m-j|   } Moore: {(k,m) : |k-i|  , |m-j|   }

(Outer) Totalistic Rules Assume k states, numbered 0,…,k-1 Let T = sum of states in neighborhood except for center cell (0  T   (k-1)r, r = # cells) A rule based only on the value T is an (outer) totalistic rule ( k [(k-1)r+1] possible rules) Totalistic rules fewer and easier to execute

Terminology (k=2) Two states: 0 and 1, dead and living Rule describes - birth (going from 0 to 1) - survival (going from 1 to 1) - death (going from 1 to 0) For Moore(  ) there are only 2 8 =256 possible Legal rules Usually: take 0  v 1  v 2  v 3  v 4  k-1, then birth for v 2  T   v 3 survival for v 1  T   v 4

The Game of Life Developed by John H. Conway in late 60s and popularized in Scientific American in 1979 by Martin Gardner Rules: Birth if T=3 Survival if 2  T   3 Many interesting questions: Do patterns stay bounded? No Maximum density pattern Longest cycle for periodic pattern Is it a Universal Touring Machine? Yes

Examples

Modifications to Life Life is a Universal Turing Machine so extra complexity is not needed, but is used in using Life to model physical systems Change rules by expanding birth range and survival range Add states representing ‘young’ and ‘old’ states Add states representing other species

Lab Time Explore Life! Assignment is to find maximum density pattern for a 10x10 grid Work on projects

Project Info Give supply list to me or Laura today Presentations should be mins. (If PowerPoint, mail it to me) Wednesday is a project workday 9-12 work in 209B, 309A, 309B or Library; I’ll be in my office Ayres 312B 1:30-? Work in 104 or computer lab You must check in sometime on Wed.