The rules for 1D CA.
A cell can be only in the states of 0 or 1. For a cell C with only 2 neighbors (L and R) we list all the possible combinations of states (configurations) written as <L|C|R> at a particular time step in the following order in n:
<1|1|1> <1|1|0> <1|0|1> <1|0|0> <0|1|1> <0|1|0> <0|0|1> <0|0|0>
where n= 7
6 5
4 3 2 1
0 .
The rule specifies whether C=1 or 0 in the next time step for each of the 8 possible
preceding configurations. For example, if our rule is that C=1 only when its previous
only neighbor with 1 was on the Left (n=6 and n=4) or its only neighbor with
1 was on the Right (n=3
and n=1) and C =0 for all other configurations, we can write the rule by expressing
the value of C for each value of n in descending order (01011010). The code used
is to treat this pattern as an 8 bit binary number and the code for the rule
is its decimal equivalent. In our example, this would be 2^6+2^4+2^3+2^1=64+16+8+2=90.
So, our example corresponds to rule 90. There are a maximum of 2^8 = 256 rules for
1D CA.
Working backwards, rule 110 would correspond to 2^6+2^5+2^3+2^2+2^1=64+32+8+4+2=110,
or the pattern
(01101110). This means that C will change its state if both L and R =1 (n=7
and 5), keep its state if both L and R =0 (n=2 and 0) or L=1 (n=6 and 4),
and will always be 1 when only R is 1 (n=3 and 1).
Starting from a single C=1 cell at the top and applying the rules as we step down
through the pattern (each line downward corresponds to a new time step) we get the
following patterns for rules 90 and 110 when C=1 is a color and C=0 is white.
1D Rule 90 1D Rule 110
The rules for 2D CA.