Small Python class for Lindenmayer Systems

Question

L-systems are basically rules for recursively rewriting a string, which can be used to characterize e.g. some fractal and plant growth.

I wrote a small class to represent deterministic L-systems and used it for two examples. Any comments would be greatly appreciated, especially about the class design, the structure of the second example, and how to make things more pythonic. I'm new to Python and don't have any training in "grammars", this is just a hobby.

The class LSystem.py:

class LSystem:
    """ Lindenmayer System
    LSystem( alphabet, axiom )
      axiom: starting "string", a list of Symbols
      rules: dictionary with rules governing how each Symbol evolves,
             keys are Symbols and values are lists of Symbols
    """
    def __init__(self, axiom, rules):
        self.axiom = axiom
        self.rules = rules

    """ Evaluate system by recursively applying the rules on the axiom """
    def evaluate(self,depth):
        for symbol in self.axiom:
            self.evaluate_symbol( symbol, depth )

    """ Recursively apply the production rules to a symbol """
    def evaluate_symbol( self, symbol, depth ):
        if depth <= 0 or symbol not in self.rules:
            symbol.leaf_function()
        else:
            for produced_symbol in self.rules[symbol]:
                self.evaluate_symbol( produced_symbol, depth - 1 )


class Symbol:
    """ Symbol in an L-system alphabet
    Symbol( leaf_function )
      leaf_function: Function run when the symbol is evaluated at the final
                     recursion depth. Could e.g. output a symbol or draw smth.
    """
    def __init__(self, leaf_function ):
        self.leaf_function = leaf_function

Example: Algae growth (example 1 from the wikipedia article)

import LSystem

# define symbols. their "leaf function" is to print themselves.
A = LSystem.Symbol( lambda:print('A',end='') )
B = LSystem.Symbol( lambda:print('B',end='') )
# define system
algae_system = LSystem.LSystem(
    axiom = [A],
    rules = { A: [A,B], B: [A] }
)
# run system
algae_system.evaluate(4) # prints "ABAABABA"

Example: Draw a Koch snowflake (as in my previous question)

import LSystem
import pygame
from math import pi, sin, cos

# some constants 
WINDOW_SIZE = [300,300]
LINE_WIDTH = 1
LINE_LENGTH = 1

# global variables for "turtle drawing"
# maybe I should pass around a turtle/cursor object instead?
turtle_angle = 0
turtle_x = 0
turtle_y = WINDOW_SIZE[1]*3/4

# define drawing functions used to draw the Koch snowflake
def draw_forward():
    global turtle_angle, turtle_x, turtle_y
    start = [turtle_x, turtle_y]
    turtle_x += LINE_LENGTH * cos(turtle_angle)
    turtle_y += LINE_LENGTH * sin(turtle_angle)
    end = [turtle_x, turtle_y ]
    pygame.draw.line(window, pygame.Color('black'), start, end, LINE_WIDTH )
def turn_left():
    global turtle_angle
    turtle_angle += pi/3
def turn_right():
    global turtle_angle
    turtle_angle -= pi/3

# symbols in the L-system
Line = LSystem.Symbol( draw_forward )
Left = LSystem.Symbol( turn_left )
Right = LSystem.Symbol( turn_right )

# L-system axiom and rules
koch_curve_system = LSystem.LSystem(
    axiom = [ Line, Right, Right, Line, Right, Right, Line ],
    rules = { Line: [ Line, Left, Line, Right, Right, Line, Left, Line ] }
)

# init pygame
pygame.init()
window = pygame.display.set_mode(WINDOW_SIZE)
window.fill(pygame.Color('white'))

# evaluate the L-system, which draws the Koch snowflake
# (recursion depth was chosen manually to fit window size and line length)
koch_curve_system.evaluate(5)

# display
pygame.display.flip()

# wait for the user to exit
while pygame.event.wait().type != pygame.QUIT:
    1

Answer 1

This is a neat implementation of Lindenmayer systems. I have some suggestions for simplifying and organizing the code.

1. The docstring for a method or a function comes after the def line (not before, as in the code here). So you need something like:

   def evaluate(self, depth):
       """Evaluate system by recursively applying the rules on the axiom."""
       for symbol in self.axiom:
           self.evaluate_symbol(symbol, depth)
   

   and then you can use the help function from the interactive interpreter:

   >>> help(LSystem.evaluate)
   Help on function evaluate in module LSystem:
   
   evaluate(self, depth)
       Evaluate system by recursively applying the rules on the axiom.
   

2. The Symbol class is redundant — it only has one attribute, and doesn't have any methods other than the constructor. Instead of constructing Symbol objects, you could just use functions:

   def A():
       print('A', end='')
   
   def B():
       print('B', end='')
   

   and instead of calling symbol.leaf_function(), you could just call symbol().

   In the Koch example, you already have functions so you can just omit the construction of the Symbol objects and write:

   koch_curve_system = LSystem(
       axiom = [draw_forward, turn_right, turn_right, draw_forward, turn_right,
                turn_right, draw_forward],
       rules = {
           draw_forward: [draw_forward, turn_left, draw_forward,
                          turn_right, turn_right, draw_forward, turn_left,
                          draw_forward],
       }
   )
   

   Alternatively, you could rename the functions and leave the definition of the system unchanged.

3. The code in evaluate is very similar to the code in evaluate_symbol. This suggests that it would result in simpler code if you described the Lindemayer system in a different way, giving an initial symbol instead of an initial list of symbols. (And possibly giving an extra rule mapping the initial symbol to a list.)

   If you try this, then you'll find that the LSystem class is redundant too: the only thing you can do with it is to call its evaluate method, so you might as well just write it as a function:

   def evaluate_lsystem(symbol, rules, depth):
       """Evaluate a Lindenmayer system.
   
       symbol: initial symbol.
       rules: rules for evolution of the system, in the form of a
           dictionary mapping a symbol to a list of symbols. Symbols
           should be represented as functions taking no arguments.
       depth: depth at which to call the symbols.
   
       """
       if depth <= 0 or symbol not in rules:
           symbol()
       else:
           for produced_symbol in rules[symbol]:
               evaluate_lsystem(produced_symbol, rules, depth - 1)
   

   Then the algae example becomes:

   evaluate_lsystem(A, {A: [A, B], B: [A]}, 4)
   

4. In the snowflake example, there is persistent shared state (the position and heading of the turtle). When you have persistent shared state it makes sense to define a class, something like this:

   class Turtle:
       """A drawing context with a position and a heading."""
       angle = 0
       x = 0
       y = WINDOW_SIZE[1]*3/4
   
       def forward(self, distance):
           """Move forward by distance."""
           start = [self.x, self.y]
           self.x += distance * cos(self.angle)
           self.y += distance * sin(self.angle)
           end = [self.x, self.y ]
           pygame.draw.line(window, LINE_COLOR, start, end, LINE_WIDTH)
   
       def turn(self, angle):
           """Turn left by angle."""
           self.angle += angle
   

   and then:

   turtle = Turtle()
   forward = lambda: turtle.forward(1)
   left = lambda: turtle.turn(pi/3)
   right = lambda: turtle.turn(-pi/3)
   initial = lambda: None
   rules = {
       initial: [forward, right, right, forward, right, right, forward],
       forward: [forward, left, forward, right, right, forward, left, forward],
   }
   evaluate_lsystem(initial, rules, 5)