L-Systems

Mar 27, 201517 min read

Deviation Actions

tatasz

Published: Mar 27, 2015

5.6K Views

Fractal Art Week

Today, we will chat about L-systems and what they have to do with fractals

Substitution systems

In a substitution system, we use a set of rules to transform elements of a sequence. Like this:

      Rule [1 → 0 and 0 → 11]
      Starting sequence: 01
      Step 1: 110
      Step 2: 0011
      Step 3: 111100
      Step 4: 00001111
      Step 5: 111111110000
      ...

A Lindenmayer system (L-system) is a string rewriting system, a substitution system that works with strings. It consists of an "alphabet" of symbols, rules (which expand symbols into larger strings - in the example above 1 → 0 and 0 → 11), an "axiom" - the starting sequence - and a way to translate the obtained strings into something visual.

L-systems can not only be used to generate fractals, but also to model many processes (plant gowth, for example - l-systems are great for making procedural trees

Koch Curve

Lets take a look at a simple example: Koch Curve.
The L-system notation for it is:

Rule: [F → F + F − F − F + F]
Start: F

Where F means "draw forward", + means "turn left 90°", and − means "turn right 90°".

Start: F

Step 1: F → F + F − F − F + F

Step 2: 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

You can see that, right now, it is actually not "substituting", but drawing forward. So the curve is not fitting a given shape, but expanding it at each step. In fractaling, though, its usually handier to actually replace elements.

For this, you would need to scale elements down - so the length of F+F−F−F+F is the same as the length of F from the previous step. With replacement, the steps above would look like:

Start

Step 1

Step 2

And 10 steps later...

In Apophysis

This second "replacement" way can be easily implemented in Apophysis (and other IFS software).
For example, the Koch Curve: our replacement rule is F → F + F − F − F + F. We will need 5 transforms, since we have 5 elemets replacing the original one on each step. As you can see on the picture above, the scale will be 1/3: the sequence F + F − F − F + F has a length 3F, and we need to scale it down so it is length F.

Now, time to position the transforms (all scaled down to 0.33333 of the original size).

leave it at the origin
Move to the right by 1/3 (length of the first F) and do the "+" - rotate by 90° CCW
Move to the right by 1/3 (first transform) and move up by 1/3 (second transform). The rotation is a "+" then "−", which cancel themselves - so no rotation at all
1/3 right, 1/3 up, 1/3 right. For rotation, we have a "+ − −" which gives a "−": 90° CW
1/3 right, 1/3 up, 1/3 right and 1/3 down - so 2/3 right overall. Rotation: "+ − − +"

Basically, you draw 1 step of the L-System rule, with transforms. Compare the picture below (F → F + F − F − F + F step)