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In above examples, each point can take the values 0, 1, 2, or 3,
corresponding respectively to the colors black, red, green, and blue.
The initial state (i.e. the top line) is chosen randomly. Then, the
cells evolve from top to bottom according to a set of rules, each line
representing a step of evolution.
How many rules are possible? If we consider only nearest neighbour
interaction, and four colors, we have 4x4x4 = 64 different patterns
possible of three cells. A rule of evolution is a function that
relates each of these 64 patterns to one color. There are
4^64 patterns possible : it is a huge number, approximately
34000000000000000000000000000000000...
A subset of these rules are those that satisfy the following two
conditions: (1) the rule depends only on the sum of the
values of the cell and its neighbours; (2) if the sum is zero, the
resulting value is zero. Even thus, there are 9 possible values of
the sum for four colors and nearest neighbour interaction, which
yeilds 4^9 = 262144 possible patterns.
Have fun exploring these patterns with the calculator below. You can
try to create patterns that give vertical lines, horizontal lines,
triangles, or tree-like components. Puzzle: see if you can reproduce
the rules for above examples!
Usage: Click on the compute button to compute a pattern. When random
is selected, the rule is randomly generated at each computation. If
continuous is selected, a new pattern is generated every 2 seconds. The
initial condition, i.e. the top line of each pattern, is always randomly
generated at each computation. In continuous mode, the computation can be
stopped at any time by deselecting the continuous checkbox. If continuous
is selected, and random is not, patterns will be periodically generated
with a constant rule, but different initial conditions.
The parameter "Extent" gives the number of neighbours on each side
that are included in the sum. The parameter "Nb shades" gives the number
of colors. When random is not selected, it is possible to edit the rule
on the right side of the screen.
Have fun and luck in your hunt. If you find a pretty pattern,
please send me its rule,
I will be happy to see it.
Now continue with the game of life.
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This page was created on 5/22/2001 by
Mite.One-dimensional cellular automata
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Copyright May 2001 by Mite.