James Adam Buckland

2015-08-06 Generating Fractals with Lindenmayer Systems graphics javascript math

**L-systems** are formal grammar structures used in the study of both botany and
mathematics, usually to simulate iterative or recursive structures, such as
algae or fractals. In this case, we employ them to render fractals.

As noted, in the Wikipedia article on L-systems (from which much of this math is cribbed), an L-system can be represented by the tuple \(\v G = (V, \omega, P)\)

- where $\v G$ is the
*alphabet*, - $\omega$ is the
*seed*, the first string, and - $P$ is the set of
*production rules*, by which the current string is replaced by definitions from the alphabet.

Letâ€™s take the famous Dragon Curve for example. Here,

- $\v G = [X, Y]$,
- $\omega = FX$, and
- $P = (X \longrightarrow X+YF+,\quad Y \longrightarrow -FX-Y)$.

In this example, $X$ and $Y$ are variables, which get replaced iteratively with
the definitions in $P$; $+$ and $-$ are *right* and *left* turns, respectively,
and $F$ means *go forwards*.

At a depth of 0, we have only the initial string $\omega$:

$FX$

At a depth of 1, we get

$FX \longrightarrow F[X]$ $\phantom{FX} \longrightarrow F[X+YF+]$ $\phantom{FX} \longrightarrow FX+YF+$

At a depth of 2, we get

$FX+YF+ \longrightarrow F[X]+[Y]F+$ $\phantom{FX+YF+} \longrightarrow F[X+YF+]+[-FX-Y]F+$ $\phantom{FX+YF+} \longrightarrow FX+YF++-FX-YF+$

You can see where this is going.

The dragon curve takes a number of iterations to get going, but many of the other curves show high levels of self-similarity at a depth of only 1 or 2. It depends on the rewriting rules.

Josh Mermelstein and I wrote a very simple
Python script which can parse these sorts of string rules, stored in a
dictionary (later, a Javascript object). It then iteratively *replaces* the
necessary strings, *swaps* out those strings for an array of commands, and then
*evaluates* each of those commands. Using Turtle
Graphics, Python then drew that
pattern on-screen. The webapp is simply a javascript port of the same code, with
the ability to modify the constants and redraw the fractal in real time.

```
from turtle import *
//recursively rewrites $str to a given $depth using an $alphabet
def rewrite(str, alphabet, depth):
if depth == 0:
steps = ''
for char in str:
if char in alphabet:
steps += alphabet[char]
else:
steps += char
return steps
else:
return rewrite(rewrite(str, alphabet, 0), alphabet, depth-1)
//translates a series of $steps into commands using an $alphabet
def translate(steps, alphabet):
commands = []
for char in steps:
commands.append(alphabet[char])
return commands
if __name__ == '__main__':
T = Turtle()
T.speed("fastest")
segmentLength = 10
depth = 3
//the entire fractal is here
rewritingRules = {
'L' : '+RF-LFL-FR+' ,
'R' : '-LF+RFR+FL-'
}
alphabet = {
'L' : '' ,
'R' : '' ,
'+' : 'T.lt(90)' ,
'-' : 'T.rt(90)' ,
'F' : 'T.fd(segmentLength)'
}
seed = 'L'
steps = rewrite(seed, rewritingRules, depth)
instructions = translate(steps, alphabet)
for item in instructions:
if item:
eval(item)
```