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Notes for: The World of Simple Programs | Substitution Systems

*History [of substitution systems]

In their various representations, 1D substitution systems have been invented independently many times for many different purposes. (For the history of fractals and 2D substitution systems see page 934.) Viewed as generators of sequences with certain combinatorial properties, substitution systems such as example (b) on page 83 appeared in the work of Axel Thue in 1906. (Thue's stated purpose in this work was to develop the science of logic by finding difficult problems with possible connections to number theory.) The sequence of example (b) was rediscovered by Marston Morse in 1917 in connection with his development of symbolic dynamics--and in finding what could happen in discrete approximations to continuous systems. Studies of general neighbor-independent substitution systems (sometimes under such names as sequence homomorphisms, iterated morphisms and uniform tag systems) have continued in this context to this day. In addition, particularly since the 1980s, they have been studied in the context of formal language theory and the so-called combinatorics of words. (Period-doubling phenomena also led to contact with physics starting in the late 1970s.)

Independent of work in symbolic dynamics, substitution systems viewed as generators of sequences were reinvented in 1968 by Aristid Lindenmayer under the name of L systems for the purpose of constructing models of branching plants (see page 1005). So-called 0L systems correspond to my neighbor-independent substitution systems; 1L systems correspond to the neighbor-dependent substitution systems on page 85. Work on L systems has proceeded along two quite different lines: modelling specific plant systems, and investigating general computational capabilities. In the mid-1980s, particularly through the work of Alvy Ray Smith, L systems became widely used for realistic renderings of plants in computer graphics.

The idea of constructing abstract trees such as family trees according to definite rules presumably goes back to antiquity. The tree representation of rule (c) from page 83 was for example probably drawn by Leonardo Fibonacci in 1202.

The first six levels of the specific pattern in example (a) on page 83 correspond exactly to the segregation diagram for the I Ching that arose in China as early as 2000 BC. Black regions represent yin and white ones yang. The elements on level six correspond to the 64 hexagrams of the I Ching. At what time the segregation diagram was first drawn is not clear, but it was almost certainly before 1000 AD, and in the 1600s it appears to have influenced Gottfried Leibniz in his development of base 2 numbers.

Viewed in terms of digit sequences, example (d) from page 83 was discussed by Georg Cantor in 1883 in connection with his investigations of the idea of continuity. General relations between digit sequences and sequences produced by neighbor-independent substitution systems were found in the 1960s. Connections of sequences such as (c) to algebraic numbers (see page 903) arose in precursors to studies of wavelets.

Paths representing sequences from 1D substitution systems can be generated by 2D geometrical substitution systems, as on page 189. The "C" curve shown on the facing page and on page 190 was for example described by Paul Lévy in 1937, and was rediscovered as the output of a simple computer program by William Gosper in the 1960s. Paperfolding or so-called dragon curves (as shown above) were discussed by John Heighway in the mid-1960s, and were analyzed by Chandler Davis, Donald Knuth and others. These curves have the property that they eventually fill space. Space-filling curves based on slightly more complicated substitution systems were already discussed by Giuseppe Peano in 1890 and by David Hilbert in 1891 in connection with questions about the foundations of calculus.

Sequences from substitution systems have no doubt appeared over the years as incidental features of great many pieces of mathematical work. As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83, then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]]^k] is equal for i=0 and i=1 if s is a sequence of the form (b) on page 83 with length 2^m, m > k. The optimal solution to the Towers of Hanoi puzzle invented in 1883 also turns out to be an example of a substitution system sequence.


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