KGameOfLife 


The Game of LifePatterns: examplesA pattern archive can be found at http://www.radicaleye.com/lifepage/picgloss/picgloss.html Let us see what happens to a variety of simple patterns. A single organism or any pair of counters, wherever placed, will obviously vanish on the first move. A beginning pattern of three counters also dies immediately unless at least one counter has two neighbors. This screenshot shows 2 generations for different patterns. The first three [a, b, c] vanish on the second move. In connection with c it is worth noting that a single diagonal chain of counters, however long, loses its end counters on each move until the chain finally disappears. The speed a chess king moves in any direction is called by Conway (for reasons to be made clear later) the "speed of light." We say, therefore, that a diagonal chain decays at each end with the speed of light. Pattern d becomes a stable "block" (twobytwo square) on the second move. Pattern e is the simplest of what are called "flipflops" (oscillating figures of period 2). It alternates between horizontal and vertical rows of three. Conway calls it a "blinker". The next picture shows you five triplets that do not fade on the first move. (Their orientation is of course irrelevant.) The illustration above shows the life histories of the five tetrominoes (four rookwiseconnected counters). The square [a] is, as we have seen, a stilllife figure. Tetrominoes b and c reach a stable figure, called a "beehive," on the second move. Beehives are frequently produced patterns. Tetromino d becomes a beehive on the third move. Tetromino e is the most interesting of the lot. After nine moves it becomes four isolated blinkers, a flipflop called "traffic lights." The illustration above shows the 12 commonest forms of still life. The reader may enjoy experimenting with the 12 pentominoes (all patterns of five rookwiseconnected counters) to see what happens to each. He will find that six vanish before the fifth move, two quickly reach a stable pattern of seven counters and three in a short time become traffic lights. The only pentomino that does not end quickly (by vanishing, becoming stable or oscillating) is the R pentomino ["a" in the illustration at the bottom of this page]. Its fate is not yet known. Conway has tracked it for 460 moves. By then it has thrown off a number of gliders. Conway remarks: "It has left a lot of miscellaneous junk stagnating around, and has only a few small active regions, so it is not at all obvious that it will continue indefinitely. After 48 moves it has become a figure of seven counters on the left and two symmetric regions on the right which, if undisturbed, would grow into a honey farm (four beehives) and traffic lights. However, the honey farm gets eaten into pretty quickly and the four blinkers forming the traffic lights disappear one by one into the rest of a rather blotchy population." As easy exercises, the reader is invited to discover the fate of the beacon, the clock and the letter H. One of the most remarkable of Conway's discoveries is the fivecounter glider. After two moves it has shifted slightly and been reflected in a diagonal line. Geometers call this a "glide reflection"; hence the figure's name. After two more moves the glider has righted itself and moved one cell diagonally down and to the right from its initial position. We mentioned above that the speed of a chess king is called the speed of light. Conway chose the phrase because it is the highest speed at which any kind of movement can occur on the board. No pattern can replicate itself rapidly enough to move at such speed. Conway has proved that the maximum speed diagonally is a fourth the speed of light. Since the glider replicates itself in the same orientation after four moves, and has traveled one cell diagonally, one says that it glides across the field at a fourth the speed of light. Movement of a finite figure horizontally or vertically into empty space, Conway has also shown, cannot exceed half the speed of light. Can any reader find a relatively simple figure that travels at such a speed? Remember, the speed is obtained by dividing the number of moves required to replicate a figure by the number of cells it has shifted. If a figure replicates in four moves in the same orientation after traveling two unit squares horizontally or vertically, its speed will be half that of light. I shall report later on any discoveries by readers of any figures that crawl across the board in any direction at any speed, however slow. Figures that move in this way are extremely hard to find. Conway knows of only four, including the glider, which he calls "spaceships" (the glider is a "featherweight spaceship"; the others have more counters). He has asked me to keep the three heavier spaceships secret as a challenge to readers. Readers are also urged to search for periodic figures other than the ones given here. The bottom illustration on this page depicts three beautiful discoveries by Conway and his collaborators. The stable honey farm ["a" in the illustration] results after 14 moves from a horizontal row of seven counters. Since a fivebyfive block in one move produces the fourth generation of this life history, it becomes a honey farm after 11 moves. The "figure 8" [b], an oscillator found by Norton, both resembles an 8 and has a period of 8. The form c, called "pulsar CP 485672," is an oscillator with a life cycle of period 3. The state shown here has 48 counters, state two has 56 and state 3 has 72, after which the pulsar returns to 48 again. It is generated in 32 moves by a heptomino consisting of a horizontal row of five counters with one counter directly below each end counter of the row. Conway has tracked the life histories of a row of n counters through n = 20. We have already disclosed what happens through n = 4. Five counters result in traffic lights, six fade away, seven produce the honey farm, eight end with four blinkers and four blocks, nine produce two sets of traffic lights, and 10 lead to the "pentadecathlon," with a life cycle of period 15. Eleven counters produce two blinkers, 12 end with two beehives, 13 with two blinkers, 14 and 15 vanish, 16 give "big traffic lights" (eight blinkers), 17 end with four blocks, 18 and 19 fade away and 20 generate two blocks. 


Last modified February 03 2003 01:39:49.
