The Rule 30 Fault Line

Rule 30 of the elementary cellular automata (CA) was among the first rules in which Stephen Wolfram noticed the appearance of intrinsic randomness in a deterministic system. When initialized with a single black pixel there is patterned behavior down both sides of the unfolding CA which gives way to the randomly patterned center. In fact, Mathematica uses the center column of pixel values as one of its random number generators.



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Snapshot 1: If all the rows are redrawn flush to the left border the structured region becomes more obvious. In fact, this shifted version of rule 30 can be emulated directly by rule 1,048,560 in the , rule space. Rule 30 as an 8-bit binary number is 00011110, while rule 1,048,560 as a 32-bit binary number is 0000-0000-0000-1111-1111-1111-1111-0000. Each bit of rule 30 expands to 4 bits of rule 1,048,560.
Snapshot 2: The regularity of the right-hand side is simpler. After the first 8 rows, the rows can be naturally grouped in powers of 2 starting with 8 rows, then 16, then 32, then 64, and so on.
Snapshot 3: Alternate display.
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