Editing L-system (section)

==== Example 1: algae, explained ====
 n=0:               A             start (axiom/initiator)
                   / \
 n=1:             A   B           the initial single A spawned into AB by rule (A → AB), rule (B → A) couldn't be applied
                 /|     \
 n=2:           A B      A        former string AB with all rules applied, A spawned into AB again, former B turned into A
              / | |       | \
 n=3:         A B A       A B     note all A's producing a copy of themselves in the first place, then a B, which turns ...
            / | | | \     | \ \
 n=4:       A B A A B     A B A   ... into an A one generation later, starting to spawn/repeat/recurse then

The result is the sequence of [[Fibonacci word]]s. If one counts the length of each string, the [[Fibonacci sequence]] of numbers is obtained (skipping the first 1, due to the choice of axiom):

:    1  2  3  5  8  13  21  34  55 89 ...

If it is not desired to skip the first 1, axiom ''B'' can be used. That would place a ''B'' node before the topmost node (''A'') of the graph above.

For each string, if one counts the ''k''-th position from the left end of the string, the value is determined by whether a multiple of the [[golden ratio]] falls within the interval <math>(k-1, k)</math>. The ratio of A to B likewise converges to the golden mean.

This example yields the same result (in terms of the length of each string, not the sequence of ''A''s and ''B''s) if the rule (''A'' → ''AB'') is replaced with (''A'' → ''BA''), except that the strings are mirrored.

This sequence is a [[locally catenative sequence]] because <math>G(n)=G(n-1)G(n-2)</math>, where <math>G(n)</math> is the ''n''-th generation.