L-systems

LSystems.py is a module for L-systems. Everything from DOL-systems to parametric context-sensitive L-systems is covered, even stochastic production rules are supported.

The aim of this library is to provide an easy tool for L-system exploration (for example in conjunction with the book: The Algorithmic Beauty Of Plants). To this end, user input is kept very simple, while still allowing for very complex production rules.

LSystems_examples.py is a module that illustrates the use of the LSystems module. LSystems_visualise.py is a module for quick visualisations of some L-systems. LSystems_3D.py contains a 3D turtle graphics class and applies this to L-systems. As a result 3D trees can be simulated using L-systems. The code also allows for the simulation of tropisms.

What are L-systems

For a quick intro into the world of L-systems see: The Algorithmic Beauty Of Plants by Przemyslaw Prusinkiewicz and Aristid Lindenmayer. Almost all examples in the code are based on examples from this book.
online: http://algorithmicbotany.org/papers/abop/abop.pdf

How to use the LSystems module

An L-system is defined by an axiom, a set of production rules and an optional ignore statement. Axioms and production rules are made up of modules (objects containing both the letter/symbol and the parameters). The axioms, production rules and ignore statement are all entered as strings.

Formatting the axioms and production rules

General formatting rules:

• Modules are seperated by spaces
• Parameters are seperated by commas
• Arrows are replaced by question marks

An example: the axiom B(2)CA(4,4) should be entered as "B(2) C A(4,4)" (C is a module with no parameters). Production rules are of the following form: left_context<predecessor>right_context:condition?successor If a certain part is not needed in the production rule it can be left out. For example a context free rule would be of the form predecessor:condition?successor. A production rule without context and condition would be predecessor?succesor. Note that the successor is always preceded by a questionmark and the condition is always preceded by a colon.

Two examples

Koch's snowflake (simple L-system)

# Specify the L-system
axiom = "F + + F + + F"
productions = ["F?F - F + + F - F"]
nrOfIterations = 5
# Initialize
system = LSystem(axiom,productions)
# Generate the L-system generations
for j in range(0,nrOfIterations):
    gen = system.nextGeneration()

After visualisation that bit of code results in: for the use of the visualisation tools see: LSystems_examples.py

Anabaena catenula (complex L-system)

# specify the L-system
axiom = "F(1,0,900) F(4,1,900) F(1,0,900)"
productions = ["F(s,t,c):t==1 and s>=6?F(s/3*2,2,c) f(1) F(s/3,1,c)",
               "F(s,t,c):t==2 and s>=6?F(s/3,2,c) f(1) F(s/3*2,1,c)",
               "F(h,i,k)<F(s,t,c)>F(o,p,r):(s>3.9 or c>0.4) and t!=0?F(s+0.1,t,c+0.25*(k+r-3*c))",
               "F(h,i,k)<F(s,t,c)>F(o,p,r):s<3.9 and c<0.4 and t!=0?F(1,0,900)",
               "F(s,t,c):t==0 and s<=3?F(s*1.1,t,c)"]
nrOfIterations = 200
ignore         = "f ~ H"
# Initialize
system = LSystem(axiom, productions,ignore)
# Generate the L-system generations
for j in range(0,nrOfIterations):
    gen = system.nextGeneration()

After visualisation that bit of code results in: on how to use the visualisation tools see: LSystems_examples.py

Stochastic production rules

To enter a stochastic production rule the following form is used: left_context<predecessor>right_context:condition?prob1;successor1;prob2;successor2 etc.

An example

Suppose we want to replace F with a 33 percent chance by F[+F]F[-F]F, with a 33 percent chance by F[+F]F and with a 34 percent chance by F[-F]F. We would enter this as:

productions = ["F?0.33;F [ + F ] F [ - F ] F;0.33;F [ + F ] F;0.34;F [ - F ] F"]

Summary of special symbols

For formatting the production rules the following symbols are used:

• : precedes the condition
• ? precedes the successor
• , seperates parameters
• ; seperates probabilities and successors (in stochastic production rules)
• space seperates modules

Bonus

With LSystems_3D.py 3D trees and shrubs can be simulated via L-systems and viewed using VPython. The code allows for tropisms (movement of plants, e.g. the movement towards light). Two possible results are shown below.

A shrub with leaves

A tree that grows towards the light (phototropism)

Problems with turtle graphics

Try changing using_IDLE = True to using_IDLE = False in the turtle.cfg file.

Acknowledgements

This project uses two modules from other authors. They are included here for completeness. It concerns the module graphics.py written by John Zelle and the module py-expression-eval.py written by Vera Mazhuga