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The width of the pattern obtained from rule 225 increases like the square root of the number of steps.

[The word] "calculus"
It is an irony of language that the word "calculus" now associated with continuous systems comes from the Latin word which means a small pebble of the kind used for doing discrete calculations (same root as "calcium").

When α is a square root, then as discussed in the previous section , the continued fraction representation is purely repetitive,
Curves obtained by adding together various sine functions.

Square root of rule 30
Although rule 30 cannot apparently be decomposed into other k = 2 , r = 1 cellular automata, it can be viewed as the square of the k = 3 , r = 1/2 cellular automata with rule numbers 11736, 11739 and 11742.

Its boundary has fractal dimension 2 Log[2, Root[2 + # 2 - # 3 , 1]] ≃ 1.52 .

For the other rules on page 952 :
d[{1, 1, 0, 1, 0}] = Log[2, Root[4 + 2 # - 2 # 2 - 3 # 3 + # 4 &, 2]] ≃ 1.72
d[{1, 1, 0, 1, 1}] = Log[2, Root[-4 + 4 # + # 2 - 4 # 3 + # 4 &, 2]] ≃ 1.80
Other cases include (see page 870 ):
d[{1, 0, 1}, k] = 1 + Log[k, (k + 1)/2]
d[{1, 1, 1}, 3] = Log[3, 6] ≃ 1.63
d[{1, 1, 1}, 5] = Log[5, 19] ≃ 1.83
d[{1, 1, 1}, 7] = Log[7, Root[-27136 + 23280 # - 7288 # 2 + 1008 # 3 - 59 # 4 + # 5 & , 1]] ≃ 1.85

The picture at the top of the facing page shows an example of a procedure for generating the base 2 digit sequence for the square root of a given number n .

But when α is not a square root the pattern can be more complicated.

Exceptions are known to include so-called Pisot numbers such as GoldenRatio , √ 2 + 1 and Root[# 3 - # - 1 &, 1] (the numerically smallest of all Pisot numbers) for which Mod[h n , 1] becomes 0 or 1 for large n .

The length of the sequence at the n th step grows like λ n , where λ ≃ 1.3 is the root of a degree 71 polynomial, corresponding to the largest eigenvalue of the transition matrix for the substitution system.