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First some background for those not yet in the know about celluar automata, they're a mathematical system for constructing graphs of cells. A cell is the smallest element of the graph, like a pixel is of an image. Cells can have only two values, ON (M in my illustration) or OFF (space). The graphing process starts at the top row and progresses row by row downward. At each row you start at the left and move to the right, determining the state of each cell in that row in turn. You make the determination based on the states of the three neighboring cells from the previous row.

The first row has to be seeded to start the process going, (usually the middle cell of the row is turned ON, or the entire row is randomized).

Here is a diagram of physical correlation of the input and output cells:
X X X
  Y

X represents an input cell, and Y represents an output cell. The "rule" determines what set of inputs corresponds to what set of outputs.

Here is Rule 30's correspondence of input to output:
1 indicates an ON cell, 0 indicates an OFF cell.

111 110 101 100 011 010 001 000
--- --- --- --- --- --- --- ---
 0   0   0   1   1   1   1   0 

The rule number comes from collecting the set of outputs as a binary number, and then converting it to decimal. As I have shown in the previous diagram, rule 30 is 00011110. If you convert 00011110 to decimal, you get 30. The rule has to have an output for every different set of inputs. Since there are 3 binary inputs to this rule, there are 8 possible input cases. So the rule has to have 8 bits to describe the output for each of those cases, one for each. With only 3 input cells, this only amounts to 256 possible rules. But with 4 and 5, there are a huge amount. These concepts are more thoroughly explained at cellular automaton, one-dimensional cellular automaton, and Wolfram's classes of cellular automata, and with a greater mathematical bias.

So without further adieu, here's rule 30 in all its glory, one of the 256 3bit rules. If you look at the large white gaps, they form triangles, which are common in CA. This CA is valued for its chaotic nature (on the columns) and therefore has applications in random number generation. Besides mathematical value, they're very interesting to look at (preferably in pixels, not ascii). I seeded it with one set cell in the middle of the first row. The number in the left column indicates the iteration count.

00                                    M
                                     MMM
                                    MM  M
                                   MM MMMM
                                  MM  M   M
05                               MM MMMM MMM
                                MM  M    M  M
                               MM MMMM  MMMMMM
                              MM  M   MMM     M
                             MM MMMM MM  M   MMM
10                          MM  M    M MMMM MM  M
                           MM MMMM  MM M    M MMMM
                          MM  M   MMM  MM  MM M   M
                         MM MMMM MM  MMM MMM  MM MMM
                        MM  M    M MMM   M  MMM  M  M
15                     MM MMMM  MM M  M MMMMM  MMMMMMM
                      MM  M   MMM  MMMM M    MMM      M
                     MM MMMM MM  MMM    MM  MM  M    MMM
                    MM  M    M MMM  M  MM MMM MMMM  MM  M
                   MM MMMM  MM M  MMMMMM  M   M   MMM MMMM
20                MM  M   MMM  MMMM     MMMM MMM MM   M   M
                 MM MMMM MM  MMM   M   MM    M   M M MMM MMM
                MM  M    M MMM  M MMM MM M  MMM MM M M   M  M
               MM MMMM  MM M  MMM M   M  MMMM   M  M MM MMMMMM
              MM  M   MMM  MMMM   MM MMMMM   M MMMMM M  M     M
25           MM MMMM MM  MMM   M MM  M    M MM M     MMMMM   MMM
            MM  M    M MMM  M MM M MMMM  MM M  MM   MM    M MM  M
           MM MMMM  MM M  MMM M  M M   MMM  MMMM M MM M  MM M MMMM
          MM  M   MMM  MMMM   MMMM MM MM  MMM    M M  MMMM  M M   M
         MM MMMM MM  MMM   M MM    M  M MMM  M  MM MMMM   MMM MM MMM
30      MM  M    M MMM  M MM M M  MMMMM M  MMMMMM  M   M MM   M  M  M
       MM MMMM  MM M  MMM M  M MMMM     MMMM     MMMM MM M M MMMMMMMMM
      MM  M   MMM  MMMM   MMMM M   M   MM   M   MM    M  M M M        M
     MM MMMM MM  MMM   M MM    MM MMM MM M MMM MM M  MMMMM M MM      MMM
    MM  M    M MMM  M MM M M  MM  M   M  M M   M  MMMM     M M M    MM  M
35 MM MMMM  MM M  MMM M  M MMMM MMMM MMMMM MM MMMMM   M   MM M MM  MM MMMM

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