L-system Drawings

Using Lindenmayer systems and Python turtle graphics to create art.

                     

Table of Contents

• Brief Introduction to L-systems
  • DOL-systems
  • Drawing L-systems
• Setup
  • For Linux
  • For MacOS
  • After Installing Initial Requirements
• Viewing the L-system Drawings
• Resources

Brief Introduction to L-systems

Lindenmayer systems (L-systems) are a string-rewriting process. They use a set of rules to recursively modify a string of symbols, transforming the initial string into something more complex. They are often applied in computer graphics, where they can be used to generate models of fractals and plants.

DOL-systems

The simplest type of L-system is a DOL-system. They are deterministic (there is exactly one way to transform each letter) and context-free (symbols from the preceding string are rewritten regardless of their position).

To better understand this system, consider this string of characters (symbols):

AB+A

This initial string is known as the axiom, where each letter corresponds to a rewriting rule. In this case, let the rules be:

A → A-B
B → AB

After three iterations of this system, AB+A would be rewritten as the following:

Iteration	String
1	A-BAB+A-B
2	A-B-ABA-BAB+A-B-AB
3	A-B-AB-A-BABA-B-ABA-BAB+A-B-AB-A-BAB

Drawing L-systems

By defining each symbol with an action, L-systems can be graphed. In particular, this project uses Python turtle graphics to draw two DOL-systems. Both use symbols with the following geometric interpretations:

Symbol	Command
F	Move forward by drawing a line of a specified length
A-Z (excluding F)	Do nothing
+	Turn right by a specified degree
-	Turn left by a specified degree
[	Push the current position and direction of the pen/turtle onto a stack
]	Pop the previous position and direction of the pen/turtle from a stack

Setup

For Linux

If Python has not been previously installed, run the following:

$ sudo apt install python3.9
$ python3.9 --version

For MacOS

If Homebrew has not been previously installed, follow the instructions listed here.

If Python has not been previously installed, run the following:

$ brew install python@3.9
$ python3.9 --version

After Installing Initial Requirements

Clone this repository:

$ git clone https://github.com/miaisakovic/l-system-drawings.git

When asked to enter credentials, input your username and personal access token.

Viewing the L-system Drawings

To view the Sierpiński arrowhead curve, run the following command:

$ python3.9 <relative path to sierpinski_arrowhead_curve.py>

To view a drawing generated by an L-system (which was provided by LSystemBot 2.0), run the following command:

$ python3.9 <relative path to l_system.py>

Resources

• Bourke, P. (1991). L-System User Notes.
• Lindenmayer, A. & Prusinkiewicz, P. (1990). The Algorithmic Beauty of Plants. Springer.
• Santell, J. (2019). L-systems.