Governor’s School for the Sciences

1. Governor’s School for the Sciences Mathematics Day 14

2. MOTD: SrinivasaRamanujan • 1887 to 1920 (India) • Worked in analytic number theory • Self-educated genius

3. 1D Results

4. Top 9 Scores • Jennifer 442 • Matt 444 • Steve 453 • Sam 456 • Charlie F 459 • Michelle 469 • Stuart 484 • Charlie W 542 • Austin 549

5. 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.

6. 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|r} Moore: {(k,m) : |k-i|r, |m-j|r}

7. (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

8. 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(r=1) there are only 28=256 possible Legal rules • Usually: take 0v1v2v3v4k-1, then birth for v2 T v3survival for v1 T v4

9. 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

10. Examples

11. 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

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

13. Project Info • Give supply list to me or Laura today • Presentations should be 12-15 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.