Overview Cellular automaton

one way simulate two-dimensional cellular automaton infinite sheet of graph paper along set of rules cells follow. each square called cell , each cell has 2 possible states, black , white. neighborhood of cell nearby, adjacent, cells. 2 common types of neighborhoods von neumann neighborhood , moore neighborhood. former, named after founding cellular automaton theorist, consists of 4 orthogonally adjacent cells. latter includes von neumann neighborhood 4 remaining cells surrounding cell state calculated. such cell , moore neighborhood, there 512 (= 2) possible patterns. each of 512 possible patterns, rule table state whether center cell black or white on next time interval. conway s game of life popular version of model. common neighborhood type extended von neumann neighborhood, includes 2 closest cells in each orthogonal direction, total of eight. general equation such system of rules k, k number of possible states cell, , s number of neighboring cells (including cell calculated itself) used determine cell s next state. thus, in 2 dimensional system moore neighborhood, total number of automata possible 2, or 7154134000000000000♠1.34×10.

it assumed every cell in universe starts in same state, except finite number of cells in other states; assignment of state values called configuration. more generally, assumed universe starts out covered periodic pattern, , finite number of cells violate pattern. latter assumption common in one-dimensional cellular automata.

a torus, toroidal shape

cellular automata simulated on finite grid rather infinite one. in 2 dimensions, universe rectangle instead of infinite plane. obvious problem finite grids how handle cells on edges. how handled affect values of cells in grid. 1 possible method allow values in cells remain constant. method define neighborhoods differently these cells. 1 have fewer neighbors, 1 have define new rules cells located on edges. these cells handled toroidal arrangement: when 1 goes off top, 1 comes in @ corresponding position on bottom, , when 1 goes off left, 1 comes in on right. (this simulates infinite periodic tiling, , in field of partial differential equations referred periodic boundary conditions.) can visualized taping left , right edges of rectangle form tube, taping top , bottom edges of tube form torus (doughnut shape). universes of other dimensions handled similarly. solves boundary problems neighborhoods, advantage is programmable using modular arithmetic functions. example, in 1-dimensional cellular automaton examples below, neighborhood of cell xi {xi−1, xi, xi+1}, t time step (vertical), , index (horizontal) in 1 generation.