Lindenmayer's Algae and the Three-Line Cut

designcraftnaturealgorithmsgrammars

Aristid Lindenmayer was a biologist, not a computer scientist. In 1968 he wanted a compact way to describe how algae and simple plants grow — cell dividing into cell — and landed on something that turned out to be a formal grammar: start with a symbol, replace every symbol at once by fixed rules, repeat. His first example modelled a filament of blue-green algae with two rules. A mature cell A becomes a mature cell followed by a young one (A→AB); a young cell B matures into A.

Run it and the strings do something quietly startling:

0  1   A
1  2   AB
2  3   ABA
3  5   ABAAB
4  8   ABAABABA
5  13  ABAABABAABAAB

The lengths are the Fibonacci numbers — which is where plants keep hiding anyway, in pine cones, sunflower spirals, and the branching of a real chrysanthemum stem.

Here it is in Clojure, where “replace everything simultaneously” is one map over a string:

(def algae {"A" "AB", "B" "A"})

(defn grow [rules axiom n]
  (nth (iterate
         #(apply str (map (fn [c] (rules (str c) (str c))) %))
         axiom)
       n))

(doseq [i (range 6)]
  (let [s (grow algae "A" i)]
    (println i (count s) s)))

Ada — the language the 1980s built for systems that must not surprise you — has no patience for strings that change length behind its back, so you declare that intent explicitly and rebuild each generation cell by cell:

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Strings.Unbounded; use Ada.Strings.Unbounded;

procedure Algae is
   S, Next : Unbounded_String := To_Unbounded_String ("A");
begin
   for Gen in 0 .. 5 loop
      Put_Line (Gen'Image & Length (S)'Image & "  " & To_String (S));
      Next := Null_Unbounded_String;
      for I in 1 .. Length (S) loop
         if Element (S, I) = 'A' then
            Append (Next, "AB");
         else
            Append (Next, "A");
         end if;
      end loop;
      S := Next;
   end loop;
end Algae;

Every L-system rule only ever adds symbols; nothing is removed. Ikebana runs the arrow the other way. You start with more branch than you need and cut back to three lines — shin, soe, hikae, roughly heaven, human, and earth — set at deliberately unequal angles so the triangle they make is scalene, never balanced. The gap between them, the ma, is counted as part of the arrangement. Same plant, two questions: the grammar asks how it multiplies, the vase asks what to take away.