S

snake

barge

boat,

long barge:

long boat

long ship

The commonest stable forms 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 Figure 129]. Conway has tracked it for 460 ticks. 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." Its fate is revealed in the addendum to this chapter.

Figure 129

- , jj. , ,, ir pentomino e i beacon vb»& , llatin cross i f

, swastika toad letter h :

.' ' ,... ! r* i b* 1pinwheel The R pentomino (a) and exercises for the reader

For such long-lived populations Conway sometimes uses a computer with a screen on which he can observe the changes. The program was written by M. J. T. Guy and S. R. Bourne. Without its help some discoveries about the game would have been difficult to make.

As easy exercises the reader is invited to discover the fate of the Latin cross ["b" in Figure 129], the swastika [c], the letter H

[d\, the beacon [e\, the clock |/], the toad [g] and the pinwheel [h]. The last three figures were discovered by Simon Norton. If the center counter of the H is moved up one cell to make an arch (Conway calls it "pi"), the change is unexpectedly drastic. The H quickly ends but pi has a long history. Not until after 173 ticks has it settled down to five blinkers, six blocks and two ponds. Conway also has tracked the life histories of all the hex-ominoes, and all but seven of the heptominoes. Some hexomi-noes enter the history of the R pentomino; for example, the pentomino becomes a hexomino on its first tick.

One of the most remarkable of Conway's discoveries is the five-counter glider shown in Figure 130. After two ticks 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 ticks the glider has righted itself and moved one cell diagonally down and to the right from its initial position. We mentioned earlier 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 ticks, 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 ticks required to replicate a figure by the number of cells it has shifted. If a figure replicates in four ticks in the same orientation after traveling two unit squares horizontally or vertically, its speed will be half that of light. Figures that move across the field by self-replication are' extremely hard to find. Conway knows of four, including the

Figure 130

Figure 130

The "glider"

The "glider"

glider, which he calls "spaceships" (the glider is a "featherweight spaceship"; the others have more counters). I will disclose their patterns in the Answer Section.

Figure 131 depicts three beautiful discoveries by Conway and his collaborators. The stable honey farm [a in Figure 131] results after 14 ticks from a horizontal row of seven counters. Since a five-by-five block in one move produces the fourth generation of this life history, it becomes a honey farm after 11 ticks. The "figure 8" [b in Figure 131], an oscillator found by Norton, both resembles an 8 and has a period of 8. The form c, in Figure 131 called "pulsar CP 48-56-72," is an oscillator with a life cycle of period 3. The state shown here has 48 counters, state two has 56 and state three has 72, after which the pulsar returns to 48 again. It is generated in 32 ticks 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 beehives 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.

Conway also investigated rows formed by sets of n adjacent counters separated by one empty cell. When n = 5 the counters interact and become interesting. Infinite rows with n= 1 or n = 2 vanish in one tick, and if n = 3 they turn into blinkers. If n = 4 the row turns into a row of beehives.

Figure 131

Figure 131

l:t honey farm i ll pulsar cp 48-56-72

l:t honey farm i ll pulsar cp 48-56-72

Three remarkable patterns, one stable and two oscillating

The 5-5 row (two sets of five counters separated by a vacant cell) generates the pulsar CP 48-56-72 in 21 ticks. The 5-5-5 ends in 42 ticks with four blocks and two blinkers. The 5-5-5-5 ends in 95 ticks with four honey farms and four blinkers, 5-5-5-5-5 terminates with a spectacular display of eight gliders and eight blinkers after 66 ticks. Then the gliders crash in pairs to become eight blocks after 86 ticks. The form 5-5-5-5-5-5 ends with four blinkers after 99 ticks, and 5-5-5-5-5-5-5, Conway remarks, "is marvelous to sit watching on the computer screen." This ultimate destiny is given in the addendum.

ANSWERS

The Latin cross dies on the fifth tick. The swastika vanishes on the sixth tick. The letter H also dies on the sixth tick. The next three figures are flip-flops: As Conway writes, "The toad pants, the clock ticks and the beacon flashes, with period 2 in every case." The pinwheel's interior rotates 90 degrees clockwise on each move, the rest of the pattern remaining stable. Periodic figures of this kind, in which a fixed outer border is required to move the interior, Conway calls "billiard-table configurations" to distinguish them from "naturally periodic" figures such as the toad, clock and beacon.

The three known spaceships (in addition to the glider, or "featherweight spaceship" are shown in Figure 132. To be precise, each becomes a spaceship in 1 tick. (The patterns in Figure 132 never recur.) All three travel horizontally to the right with half the speed of light. As they move they throw off sparks that vanish immediately as the ships continue on their way. Unescorted spaceships cannot have bodies longer than six counters without giving birth to objects that later block their

Figure 132

Lightweight (left), middleweight (center), and heavyweight (right) spaceships motion. Conway has discovered, however, that longer spaceships, which he calls "overweight" ones, can be escorted by two or more smaller ships that prevent the formation of blocking counters. Figure 133 shows a larger spaceship that can be es-

Overweight spaceship with two escorts corted by two smaller ships. Except for this same ship, lengthened by two units, longer ships require a flotilla of more than two companions. A spaceship with a body of 100 counters, Conway finds, can be escorted safely by a flotilla of 33 smaller ships.

ADDENDUM

My 1970 column on Conway's "Life" met with such an instant enthusiastic response among computer hackers around the world that their mania for exploring "Life" forms was estimated to have cost the nation millions of dollars in illicit computer time. One computer expert, whom I shall leave nameless, installed a secret switch under his desk. If one of his bosses entered the room he would press the button and switch his computer screen from its "Life" program to one of the company's projects. The next two chapters will go into more details about the game. Here I shall comment only on some of the immediate responses to two questions left open in the first column.

The troublesome R pentomino becomes a 2-tick oscillator after 1,103 ticks. Six gliders have been produced and are traveling outward. The debris left at the center [see Figure 134] consists of four blinkers, one ship, one boat, one loaf, four beehives, and eight blocks. This was first established at Case Western Reserve University by Gary Filipski and Brad Morgan, and later confirmed by scores of "Life" hackers here and abroad.

The fate of the 5-5-5-5-5-5-5 was first independently found by Robert T. Wainwright and a group of hackers at Honeywell's Computer Control Division, later by many others. The pattern stabilizes as a 2-tick oscillator after 323 ticks with four traffic lights, eight blinkers, eight loaves, eight beehives, and

R pentomino's original (black) and final (open dots) state. (Six gliders are out of sight.)

four blocks. Figure 135 reproduces a printout of the final steady state. Because symmetry cannot be lost in the history of any life form, the vertical and horizontal axes of the original symmetry are preserved in the final state. The maximum population (492 bits) is reached in generation 283, and the final population is 192.

Figure 135

Initial pattern and final state of the 5-5-5-5-5-5-5 row

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