## Life

The game of Life is a computer game, invented by John Horton Conway, and popularized by Martin Gardner in Scientific American in 1970. The game is a system of cellular automata (pronounced ah-TOM-uh-tuh, it is the plural of automaton), or artificial life. It is played on an infinite checker-type board. Computers cannot actually deal with an infinite board, so occasionally there is overflow. The object is to set up interesting initial patterns, and see how they evolve. A simple example is shown in the animation above left.

The rules of the game are these:

• A cell (one square on the board) can either be on or off.
• A cell's neighbors are those eight cells adjacent to it, including diagonally.
• Each move (tick) of the game is one generation of the organism (collection of cells).
• These rules determine which cells change in the next generation:
1. If the number of "on" neighbors is exactly 2, then the cell stays the same.
2. If the number of "on" neighbors is exactly 3, then the cell will be on.
3. Otherwise, the cell will be off.

Let me illustrate these rules by tracing the lives of four life forms, above.

1. #1 begins with three cells that are on. Each of these lives in the next generation, because the number of "on" neighbors is exactly two. And the fourth cell (completing the fourth square of the "block") turns on, as it has exactly three "on" neighbors. The resultant block of four cells then remains unchanged in subsequent generations.
2. #2 is the "blinker." The end cells die, while one cell above and below gets turned on. This then oscillates forever with a period of 2. I have animated it, below.
3. #3 dies out. First the two end cells die, and then the middle one dies as it is has no neighbors.
4. #4 begins with five cells in a straight line. It then becomes the nine cells in the square, below that. Then it becomes the sparse diamond shape, then the thicker diamond shape, then the "o" shape, then the arrangement of four "+" shapes. Then it changes into "traffic lights," which are four blinkers which oscillate with a period of two.

The examples that I animate here are fairly simple. The first one dies out, its ends moving at speed c (the speed of light, or as fast as any effect can move in the game). The second one is called a fuse, and becomes stable (there are various kinds of fuses). The first object was a kind of degenerate fuse. And the third one is called a glider, and it moves diagonally at c/4, and is the simplest spaceship. The speed c is one square per tick (generation) horizontally, vertically, or diagonally. A phenomenon moving at c in all directions spreads out in an expanding octagon.

The thirteen objects on the left are stable, never changing unless struck by some moving object. They are called still lifes. The first five are called, respectively, the block, tub, boat, beehive, and loaf. There are many other stable objects, many of which have names. The next to the last one is called the eater. After quite a few different objects collide with it (at its upper left corner), it restores itself (ending up unchanged) while the other object is absorbed into it. The last one shows that some of these can be stretched to greater length. This one is a particularly long boat.

On the left, we have an animation with four simple oscillators, all of which have periods of two. They are called the beacon, clock, blinker (which is very common indeed), and toad. Four blinkers (as seen in the final stages of the animation at the top of this article) form a traffic light. On the right, is the barber pole, which also has a period of 2 It can be stretched to any length.

Other organisms are more complicated, and even more interesting. The four objects on the left are surprisingly prolific. I won't animate any of these. The first one is the pi heptamino which lasts for 173 generations before settling down to a hodgepodge of stable objects and blinkers. The second is the gamma hexamino, and goes for over 1000 generations before settling down to stable objects, blinkers, and gliders. The third is the R-pentamino, which lasts 1103 generations before settling down to stable objects, blinkers, and gliders. These two are Methuselah objects, meaning that they last a long time. The fourth one is the acorn, another Methuselah object, which settles down to debris and gliders after over 5000 generations.

On the right is the Gosper glider gun, which oscillates with a period of 30, and spews out gliders. This means that it continually produces new matter forever, which answers the question, "Can any organism grow forever?" A glider gun can even be created by the collisions of eight gliders.

The three objects on the left are spaceships, which move toward the right at c/2, while oscillating with a period of four. Actually, the images shown here do not recur, but are spaceship precursors, and immediately become spaceships. Larger spaceships require help (attending spaceships) to keep them from breaking apart.

Below are some oscillators with periods greater than 2. They are the pulsar (period-3) which is surprisingly common in symmetrical situations, two eaters (period-3) [this animation is currently defective], another period-3 oscillator, a period-4 oscillator, the pinwheel and a similar object (both period-4), a period-5 oscillator (drawn to a smaller scale), and a period-6 oscillator (drawn to a smaller scale). The last three, which I won't animate yet, are the figure 8 (period-8), and objects of period 14 (the last 7 are upside down) and 15:

Oscillators of arbitrarily long periods can be constructed using a glider which goes back and forth between arbitrarily distant objects. Certain specific periods (such as 19) may not be possible, however.

Creating large random life forms, and then seeing what happens to them, is also interesting.

On the left is the famous Garden of Eden life form. It is apparently the smallest (?) life form which cannot arise from any other pattern. It becomes a still life after 134 generations.

It is possible to make a Turing machine (which I will write about) out of a very complicated array of life forms, using gliders as data. And therefore it is possible to simulate any digital computer (including any programs) with the game of life. One might call this a life long project.

Life may be a fruitful topic for any number of science fair projects.

There are numerous life programs out there. See Life32, a program for Windows, and Jason's Life Page. See some of the amazingly complicated life forms out there, including ones that grow exponentially.

I read once that life tends to produce symmetry. I would say that this is mostly not true, except on a small scale. It does preserve symmetry; so accidental local symmetry stays around until it is destroyed by moving objects from elsewhere. Also, symmetric objects tend to settle down into stable (dead?) pieces sooner. Asymmetry tends to be more interesting.

Note: the rules can also be stated as these:

1. If a cell is on and the number of "on" neighbors is 2 or 3, then the cell stays the same.
2. If a cell is off and the number of "on" neighbors is exactly 3, then the cell will turn on.
3. Otherwise, the cell will be off.