Governor’s School for the Sciences

MathematicsMathematicsDay 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 W 542• 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 28=256 possible Legal rules

• Usually: take 0v1v2v3v4k-1, then birth for v2 Tv3

survival for v1 Tv4

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 T3

• 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 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.