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Life
© Copyright 1998 and 2002, Jim Loy
Note: This page uses GIF animation. My apologies if your browser does
not support this feature. See the addendum below.
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:
- If the number of "on" neighbors is exactly 2, then the cell
stays the same.
- If the number of "on" neighbors is exactly 3, then the cell will
be on.
- Otherwise, the cell will be off.
Let me illustrate these rules by tracing the lives of four life forms,
above.
- #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 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 dies out. First the two end cells die, and then the middle one
dies as it is has no neighbors.
- #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:
- If a cell is on and the number of "on" neighbors is 2 or 3, then the
cell stays the same.
- If a cell is off and the number of "on" neighbors is exactly 3, then
the cell will turn on.
- Otherwise, the cell will be off.
Addendum #1:
For you folks with caveman browsers, here are a few GIFs to help you
with some of the above animations. Above left we see three generations each of
the four period-2 oscillators that I showed above (rotated 90 degrees), the
beacon, clock, blinker, and toad. To the right of that, we have the period-3
pulsar. And to the right of that we have five images of the glider, showing it
moving one square right and one down. Below we have one moment in the life of
the Gosper glider gun (with the blank squares drawn in gray). Gliders emerge
from near the center, and travel down and right:
The starting states of the other oscillators that I animated above are
these (with periods of 3, 3, 4, 4, 4, 5, and 6):
Addendum #2:
On the left, we see two generations of a period-2
oscillator that I discovered. It has probably been seen before, as it is fairly
simple. But, I don't see it on the WWW, so far.
Life is generally unpredictable; it is chaotic (see
Chaos - Sensitivity To Initial Conditions).
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