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Unformatted text preview: Chapter 9 Game of Life Conways Game of Life makes use of sparse matrices. The Game of Life was invented by John Horton Conway, a British-born mathematician who is now a professor at Princeton. The game made its public debut in the October 1970 issue of Scientific American , in the Mathematical Games column written by Martin Gardner. At the time, Gardner wrote This month we consider Conways latest brainchild, a fantastic solitaire pastime he calls life. Because of its analogies with the rise, fall and alternations of a society of living organisms, it belongs to a growing class of what are called simulation games games that resemble real-life processes. To play life you must have a fairly large checkerboard and a plentiful supply of flat counters of two colors. Of course, today we can run the simulations on our computers. The universe is an infinite, two-dimensional rectangular grid. The population is a collection of grid cells that are marked as alive . The population evolves at discrete time steps known as generations . At each step, the fate of each cell is determined by the vitality of its eight nearest neighbors and this rule: A live cell with two live neighbors, or any cell with three live neigbhors, is alive at the next step. The fascination of Conways Game of Life is that this deceptively simple rule leads to an incredible variety of patterns, puzzles, and unsolved mathematical problems just like real life. If the initial population consists of only one or two live cells, it expires in one step. If the initial population consists of three live cells then, because of rotational Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1 2 Chapter 9. Game of Life Figure 9.1. A pre-block and a block. and reflexive symmetries, there are only two different possibilities the population is either L-shaped or I-shaped. The left half of figure 9.1 shows three live cells in an L-shape. All three cells have two live neighbors, so they survive. The dead cell that they all touch has three live neighbors, so it springs to life. None of the other dead cells have enough live neighbors to come to life. So the result, after one step, is the population shown in the right half of figure 9.1. This four-cell population, known as the block , is stationary. Each of the live cells has three live neighbors and so lives on. None of the other cells can come to life. Figure 9.2. A blinker blinking. The other three-cell initial population is I-shaped. The two possible orien- tations are shown in each half of figure 9.2. At each step, two end cells die, the middle cell stays alive, and two new cells are born to give the orientation shown in the other half of the figure. If nothing disturbs it, this blinker keeps blinking forever. It repeats itself in two steps; this is known as its period ....
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