But for multiway systems where each rule p  q is accompanied by its reverse q  p , and such pairs are represented say by "AAB" ↔ "BBAA" , an equivalent operator system can immediately be obtained either from Apply[Equal, Map[Fold[#2[#1] &, x, Characters[#]] &, rules, {2}], {1}] or from (compare page 1172 ) Append[Apply[Equal, Map[(Fold[f, First[#], Rest[#]] &)[Characters[#]] &, rules, {2}], {1}], f[f[a, b], c]  f[a, f[b, c]]] where now objects like "A" and "B" are treated as constants—essentially functions with zero arguments.

Frequency of behavior [in multiway systems] Among multiway systems with randomly chosen rules, one finds about equal numbers that grow rapidly and die out completely.

Hamming distances [in networks] In the so-called loop switching method of routing messages in communications systems one lays out a network on an m -dimensional Boolean hypercube so that the distance on the hypercube (equal to Hamming distance) agrees with distance in the network.

Each patch is set up to have a roughly equal number of black and white squares.

The digits of 1/n in base b repeat with period MultiplicativeOrder[b, FixedPoint[#/GCD[#, b] &, n]] which is equal to MultiplicativeOrder[b, n] for prime n , and is at most n - 1 .

Musical chords [from sine sums] In a so-called equal temperament scale the 12 standard musical notes that make up an octave have a progression of frequencies 2 n/12 .

In the most naive economic theory, price is a reflection of value, and the value of an asset is equal to the total of all future earnings—such as dividends—which will be obtained from it, discounted for the interest that will be lost from having to wait to get these earnings.

It can be shown that if the pattern is extended sufficiently far, then any possible local arrangement of black squares will eventually appear—though not necessarily with equal frequency.

With black and white interpreted as True and False , the forms of operators shown here correspond respectively to And , Equal , Implies and Nand .

Labelling each shape and orientation with a different color, the behavior of this system can be reproduced with equal-sized squares using the rule {3  {{1, 0}, {3, 2}}, 2  {{1}, {3}}, 1  {{3, 2}}, 0  {{3}}} starting from initial condition {{3}} .