Lindenmayer System in Clojure
An L-system or Lindenmayer system is a language, which means a set of strings that is made by the application using certain rules. L-systems can be used to generate fractals such as iterated function systems.
- variables : F
- constants* : + -
- start : F
- rules : (F -> F+F-F-F+F)
Above grammar represents the Koch curve, where F means "draw forward", + means "turn left 90",and - means "turn right 90". Following is a simple implementation in Clojure to build sentences using grammars defined like this.
(ns lsystem (:use turtle)) (defn variable? [grammer symbol] (contains? (:variables grammer) symbol)) (defn apply-rules [grammer sentence] (flatten (map #(if (variable? grammer %) ((:rules grammer) %) %) sentence))) (defn l-system [grammer n] (loop[acc n sentence (:start grammer)] (if (= 0 acc) sentence (recur (dec acc) (apply-rules grammer sentence)))))
We take the axiom (start) and apply the rules n times, with each iteration we replace the variables with their corresponding rules, so the grammar above will grow such as,
- n=0: F
- n=1: F+F-F-F+F
- n=2: F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F
Grammar for the Koch curve in Clojure is represented like so,
(def koch-curve {:variables #{:F} :constants #{:+ :-} :start [:F] :rules {:F [:F :+ :F :- :F :- :F :+ :F]} :actions {:F forward :+ left :- right} :angle 90 :step 10})
There are some additions to the grammar, such as what actions will be mapped to the variables while drawing, angles for the turns and a step value to determine how much to move forward or backward.
(defn draw-system [turtle grammer sentence] (doseq [letter sentence] (let [action (letter (:actions grammer))] (cond (or (= action forward) (= action back)) (action turtle (:step grammer)) (or (= action left) (= action right)) (action turtle (:angle grammer)))))) (defn setup-turtle [turtle x y] (pen-up turtle) (right turtle 90) (go turtle x y) (pen-down turtle))
After creating a sentence for the fractal all we need to do is, iterate over the letters and command turtle to do the action that is mapped to the letter.
(def dragon-curve {:variables #{:X :Y} :constants #{:F :+ :-} :start [:F :X] :rules {:X [:X :+ :Y :F] :Y [:F :X :- :Y]} :actions {:F forward :+ left :- right} :angle 90 :step 10}) (doto (turtle 450 600) (setup-turtle -50 -200) (draw-system dragon-curve (l-system dragon-curve 10)) (show))
(def pentigree {:variables #{:F} :constants #{:+ :-} :start [:F :- :F :- :F :- :F :- :F] :rules {:F [:F :- :F :+ :+ :F :+ :F :- :F :- :F]} :actions {:F forward :+ left :- right} :angle 72 :step 10}) (doto (turtle 400 400) (setup-turtle -90 -100) (draw-system pentigree (l-system pentigree 3)) (show))
(def sierpinski-triangle {:variables #{:A :B} :constants #{:+ :-} :start [:A] :rules {:A [:B :- :A :- :B] :B [:A :+ :B :+ :A]} :actions {:A forward :B forward :+ left :- right} :angle 60 :step 10}) (doto (turtle 700 600) (setup-turtle -300 -200) (draw-system sierpinski-triangle (l-system sierpinski-triangle 6)) (show))