plainsaw

The First Row and Grouping Results

Another significant early observation was that in single-seed, three neighbor elementary automata, the rules used on the first row of the pattern are the same for all 256 rulesets. This is not to say that the active/passive patterns are the same -- because they're not. Instead, it says the same underlying rules are called upon to determine which cells in the first row are to be active or passive. The figure below shows the seed row and the rules invoked in the first row:

		1	2	4

Once I looked at the mechanics, it became obvious why this occurs:

• The cells to the left and the right of the seed cell invoke 0:000 (Rule 0) because all three of the predecessor cells are considered to be in the passive state -- a grid cell that isn't active or passive is treated as passive.
• The cell diagonally to the left of the seed invokes  1:001  because the neighboring cells are passive, passive, and active.
• The cell below the seed invokes  2:010  because the neighbors are passive, active, and passive.
• The cell diagonally to the right invokes  4:100  because the neighbors are active, passive, and passive.
• The remaining rules can not be used because they require at least two active neighboring cells; therefore it is impossible for them to be invoked by a single seed cell. This is also a more general rule -- wherever in a pattern there is a sinlge cell, only  1:001 ,  2:010 , and  4:100  can be invoked.

Once past the first row, the mix of active and passive cells gnerates a variety of patterns.

The distinction between the process of using rules and the results generated is apparent in the first row. If all four rules generate passive cells, then the first row will be white and, in fact, the whole pattern will be white. If all four rules generate active cells, the first row and the entire pattern will be entirely black.

It turns out there are the sixteen possible active/passive patterns generated by the first row of a single seed cell automaton. The three rules,  1:001 ,  2:010 , and  4:100 , generate eight possible active/passive patterns (which are just like the patterns of the eight possible rules). The fourth rule, 0:000, affects the rest of the cells in the first row. When Rule 0 is set to passive, the resuling cells are all white and when set to active, the cells are black and produce a black stripe. Below is a chart of the sixteen possible results:

Early on in my investigations, I tried various ways to organize or group patterns together based on visual appearance with little success. Using the results from the first row, however, provides an organization structure which brings the rulesets together in a thoroughly rational and coherent manner. See here for the Results Groups of the rulesets.

 

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