Creating generative art with L-systems in Python

Some experiments on the algorithmic beauty of plants

english
visualization
generative art
coding

Would it be possible to create a Python project that can generate those pencil-like drawings with random forests, with different types of trees?“, I thought (pun intended). Of course, someone already thought of that and it even has a name: algorithmic botany.

Published

Apr 2021

Creative coding is fun if you know a thing or two about programming. And if, like me, you are an utter noob when it comes to brushes and pencils, then it is even more enticing as a means to express yourself and create something just for the sake of it. It doesn’t have to be functional, but you do get to hone your hacking skills and set challenges for yourself along the way.

A fellow data scientist who is into “Art & AI” recently pointed me to the field of generative art. I have always been facinated by emergence and even more so when I came across the picture above. It is featured in this blog on how to draw a pencil-sketch-like forest with Processing. One thing led to another: I had a go at Processing in Python mode, dabbled with matplotlib for animations and read Schiffman’s the Nature of Code.

“Would it be possible to create a Python project that can generate those pencil-like drawings with random forests, with different types of trees?”, I thought (pun intended). Of course, someone already thought of that and it even has a name: algorithmic botany. Przemyslaw Prusinkiewicz and Aristid Lindenmayer wrote the seminal book on the subject in 1990, The Algorithmic Beauty of Plants. Lindenmayer invented L-systems already back in 1968. Today, household computers have enough resources to handle large-ish parallel writing systems and you could start modelling the growth of any plant using a framework like L-Py.

Although a lot has been written on the subject and there are quite a few open source libraries, they weren’t quite what I have in mind. Instead, the idea of creating generative art in Python emerged using the following components:

  1. A lightweight implementation of L-systems that also supports stochastic and parametric production rules;
  2. Integration with p5js via pyp5js as a web-native graphing engine.

The possibilities are endless and I can foresee spending way too much time on this. In this blog post, I will lay out the ideas behind this approach.

Playing with L-systems in Python

There are different kinds of L-systems, with increasing complexity. We will start with the simplest: Lindenmayer’s original L-system for modelling the growth of algae (We will jump right in, please read the article on Wikipedia if you need more context).

D0L: deterministic context-free L-system

# define L-system
AXIOM = 'A'
RULES = {'A': 'AB',
         'B': 'A'}


# initialize 
productions = [AXIOM]
iterations = 7


def generate_d0l(axiom, productions, iterations):
    def rewrite_l(word):
        productions.append(''.join([RULES.get(c) or c for c in word]))
        return productions[-1]


    if iterations == 0:
        return axiom
    else:
        return rewrite_l(generate_d0l(axiom, productions, iterations - 1))


# main
generate_d0l(AXIOM, productions, iterations)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: A
1: AB
2: ABA
3: ABAAB
4: ABAABABA
5: ABAABABAABAAB
6: ABAABABAABAABABAABABA
7: ABAABABAABAABABAABABAABAABABAABAAB

I actually found the recursive Python implementation on this introductory website on recursion. While I still struggle to apply recursion, I find it satisfying to solve coding problems this way. And even with just these twenty lines of code, we have already stumbled on the Fibonacci numbers! Have a look at the length of each successive sequence:

print([len(word) for word in productions])
[1, 2, 3, 5, 8, 13, 21, 34]

Awesome as the nature of code may be, we are still a long way off from actually generating a tree. For that, we need branching rules. To do so, basic L-systems also allow constants, which are left untouched by the production rules. It is common to add constants + and - to denote rotations when branching, and [,] for handling the locations of the branching points. For a fractal plant, the code is as follows:

AXIOM = "X"
RULES = {"X": "F+[[X]-X]-F[-FX]+X",
         "F": "FF"}
productions = [AXIOM]
generate_d0l(AXIOM, productions, iterations)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: X
1: F+[[X]-X]-F[-FX]+X
2: FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X
3: FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X
4: FFFFFFFF+[[FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFF[-FFFFFFFFFFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X
5: FFFFFFFFFFFFFFFF+[[FFFFFFFF+[[FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFF[-FFFFFFFFFFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFF+[[FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFF[-FFFFFFFFFFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFFFFFFFFFF[-FFFFFFFFFFFFFFFFFFFFFFFF+[[FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFFFFFF[-FFFFFFFFFFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]+FFFFFFFF+[[FFFF+[[FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-X]-F[-FX]+X]-FFFF[-FFFFFF+[[F+[[X]-X]-F[-FX]+X]-F+[[X]-X]-F[-FX]+X]-FF[-FFF+[[X]-X]-F[-FX]+X]+F+[[X]-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6: 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It is amazing to see how complexity increases by adding slightly longer rules. To actually draw something, we translate the letters as follows: * “F” means “draw forward” * “-” means “turn right”, using a predefined value e.g. 30° * “+” means “turn left 30°” * “X” does not correspond to any drawing action and is used to control the evolution of the curve.

The square bracket “[” corresponds to saving the current values for position and angle, which are restored when the corresponding “]” is executed, which effectively is a LIFO stack. We will go into the details of how to draw later.

Adding stochastic rules

D0L systems are completely deterministic, that is, each starting axiom yields the same result. Using this to draw a forest would result in a boring scene: all trees would be identical. To mimic mutations in nature, stochastic L-systems allow the same letter to have for different production rules with a given probability. For example, if we want three different production rules for F with equal probability

p1: F(0.33) → F[+F]F[−F]F
p2: F(0.33) → F[+F]F
p3: F(0.34) → F[−F]F

the Python implementation could look something like this:

from numpy.random import default_rng

AXIOM = "F"
RULES = {"F": ["F[+F]F[−F]F", "F[+F]F", "F[−F]F"]}
productions = [AXIOM]
iterations = 3


rng = default_rng()
def generate_stochastic(axiom, productions, iterations):
    def get_stochastic_rule(letter):
        if letter not in RULES.keys():
            return letter
        else:
            return RULES.get(letter)[rng.choice(3,1)[0]]


    def rewrite_l(word):
        productions.append(''.join([get_stochastic_rule(c) for c in word]))
        return productions[-1]


    if iterations == 0:
        return axiom
    else:
        return rewrite_l(generate_stochastic(axiom, productions, iterations - 1))


generate_stochastic(AXIOM, productions, iterations)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: F
1: F[−F]F
2: F[−F]F[−F[−F]F]F[−F]F
3: 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]F[−F]F]F[+F]F[−F]F

Here’s another run with a different result:

# show that a new run generates different result
productions = [AXIOM]
generate_stochastic(AXIOM, productions, iterations)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: F
1: F[+F]F[−F]F
2: F[+F]F[−F]F[+F[+F]F]F[−F]F[−F[+F]F[−F]F]F[+F]F[−F]F
3: 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]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]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[+F]F[−F]F

Adding parameters

In a parametric L-system, letters are allowed to have parameters, which correspond with formal parameters used in the specification of L-system productions. This can be used to model Anabaena catenula (taken from Hanan’s dissertation):

from collections import namedtuple


# parameters
F = namedtuple('F', ['s', 't', 'o'])  # parameterized cell
s = {2: "AS", 1: "BS"}  # cell size
t = {1: "A", 2: "B"}  # cell type
o = {1: "right", -1: "left"}  # cell orientation

AXIOM = [F(1, 2, 1)]
RULES = {F(2,1,1):  [F(2,1,-1), F(1,2,1)],
         F(2,1,-1): [F(1,2,-1), F(2,1,1)],
         F(1,2,1):  [F(2,1,1)],
         F(1,2,-1): [F(2,1,-1)],
        }
productions = [AXIOM]
iterations = 5


def unnest(list_, unnested_=None):
    """Utility function to unnest arbitrarily deep list of lists
    """
    if unnested_ is None:
        unnested_ = []
    for x in list_:
        if not isinstance(x, list):
            unnested_.append(x)
        if isinstance(x, list):
            unnest(x, unnested_)
    return unnested_


def generate_parametric(axiom, productions, iterations):
    def rewrite_l(word):
        productions.append(unnest([RULES.get(c) or c for c in unnest(word)]))
        return productions[-1]


    if iterations == 0:
        return [axiom]
    else:
        return rewrite_l(generate_parametric(axiom, productions, iterations - 1))


generate_parametric(AXIOM, productions, iterations)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: [F(s=1, t=2, o=1)]
1: [F(s=2, t=1, o=1)]
2: [F(s=2, t=1, o=-1), F(s=1, t=2, o=1)]
3: [F(s=1, t=2, o=-1), F(s=2, t=1, o=1), F(s=2, t=1, o=1)]
4: [F(s=2, t=1, o=-1), F(s=2, t=1, o=-1), F(s=1, t=2, o=1), F(s=2, t=1, o=-1), F(s=1, t=2, o=1)]
5: [F(s=1, t=2, o=-1), F(s=2, t=1, o=1), F(s=1, t=2, o=-1), F(s=2, t=1, o=1), F(s=2, t=1, o=1), F(s=1, t=2, o=-1), F(s=2, t=1, o=1), F(s=2, t=1, o=1)]

While the Anabaena example above only makes simple comparisons, parametric L-systems can use any expression using the parameters in combination with the arithmetic operators +, −, ∗, /; the exponentiation operator ∧, the relational operators <, >, =; the logical operators !, &, | (not, and, or) etc. For example, we could define a parameter time which increases with every iteration. This could be used to thicken the width of a branch of a tree.

Using pyp5js for Pythonic drawing on the web

When Lindenmayer devised the L-system, Logo was one of the few programming langauges available. It included turtle graphics and thus the basis of algorithmic art was born by translating the alphabet of an L-system to turtle commands. The examples from the Nature of Code have in fact implemented this in Processing. But rather than constrain yourself to the desktop, more mileage can be had by using p5js which is “… an interpretation of Processing for today’s web”. To demonstrate, take the following example:

AXIOM = 'G'
RULES = {
    'F': 'FF',
    'G': 'F-[[G]+G]+F[+FG]-G'
}
productions = [AXIOM]
generate_d0l(AXIOM, productions, iterations=5)
print(''.join([f"{i}: {p}\n" for i,p in enumerate(productions)]))
0: G
1: F-[[G]+G]+F[+FG]-G
2: FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G
3: FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G
4: FFFFFFFF-[[FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF[+FFFFFFFFFFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G
5: FFFFFFFFFFFFFFFF-[[FFFFFFFF-[[FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF[+FFFFFFFFFFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF-[[FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF[+FFFFFFFFFFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFFFFFFFFFF[+FFFFFFFFFFFFFFFFFFFFFFFF-[[FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF[+FFFFFFFFFFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFFFFFF-[[FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFFFFFF[+FFFFFFFFFFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FFFF-[[FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]+FFFF[+FFFFFF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G]-FF-[[F-[[G]+G]+F[+FG]-G]+F-[[G]+G]+F[+FG]-G]+FF[+FFF-[[G]+G]+F[+FG]-G]-F-[[G]+G]+F[+FG]-G

Summer Rizo has already implemented L-systems in p5js, available on thefractal.zone. Copying the axiom and rules, and setting the branching angle \(\theta\) to 30°, yields the drawing below. Note that the code only shows 5 iterations, while the drawing is based on more iterations (I have put the slider on maximum). When using recursion in Python you may quickly hit a limit: as I tried to execute iterations=10 on a standard VM, Deepnote threw an error IOPub data rate exceeded. Yet another challenge to read-up and use tail-call optimization in Python.

Having covered the algorithmic and technical components, we can now start bringing it all together. There are still quite a few things to experiment with in order to produce realistic sketches, such as adding noise to draw ‘sketchy’ lines, adding thickness to branches, adding shadows with brightness off-sets and composing a scene with overlaying trees. As a Pythonista I was happy to find pyp5js, which allows you to re-use the ideas of Processing in Python, and transcoding it to p5.js using Transcrypt. It covers all the methods from p5.js and gives us enough drawing tools to play with. But that’s for another post. For now, I will leave you with a generative creation from Anders Hoff which may inspire you to start creating generative art in Python, too.