Month: February 2011

In this post I will discuss some of the different patterns created by modifying the start condition in the Lindenmayer System Explorer. Clicking on any of the images will take you to the explorer tool, preloaded with the variables necessary to re-create that image.

So by now you all have seen the basic pattern which is created by the default settings in the explorer:

All well and good, but it feels incomplete; maybe a little lop-sided. Change the start condition to “F+F+F+F”, and you will see this:

How did this happen?!? Look at the angle: 90°. When you click the render button, the start condition and the grammar are run through an algorithm which creates a long string of characters. Every time an “F” is encountered, a line segment is created. Every time a “+” or “-” is encountered, the angle at which the next segment will be drawn is updated by the value in the “angle” field. “+” turns clockwise, “-” turns widdershins. So in this instance, every “+” means the next line will be drawn at a 90° angle to the previous segment. In the first example, having “F” as the starting condition drew, overall, a single quarter of a square pattern. Changing the start condition to “F+F+F+F” means that the initial 90° angle would be repeated 4 times, each at a 90° offset from the previous. 90 x 4 = 360°, which brings the line back to the start position.

This will work with any number which divides evenly into 360. Here is a 5-sided (72°) figure:

Six sides at 60°:

…and so on. As long as the starting condition and angles are correct, you can put almost anything in the grammar and use any number of iterations, and the result will still be a closed shape. Here are a few more:

In the Lindenmayer System Explorer tutorials I am posting I continually refer to “rule sets”, which are the DNA (so to speak) of the shapes which are created when the you click the “RENDER” button.

Rule sets are built as follows:
1. look at the current rule set, which in this instance is equal to the start condition “F”.
2. Look at the grammar, which in this instance is “F:F+F-F-F+F”.
3. Match the grammar character to the left of the colon to the characters in the start condition. Each time you encounter an “F” in the start condition, replace it with everything to the right of the colon in the grammar.
4. If we have not yet done this as many times as the iterator calls for, go back to (1) and repeat it with the updated rule set.

So with the default start condition and grammar, and the iterations set to “1”, the resulting rule set looks like this:

F+F-F-F+F

The Explorer simply replaced all of the “F” characters in the start condition with the entire grammar. Simple enough.

Now change the iterations to “2”, and click “RENDER“. The rule set now looks like this:

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

Instead of 9 characters, the rule set is now 49 characters long, and the resulting shape is more complex.

Changing the iterations to 3 will give you this 249-character long rule set:

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+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

…and here is 4 iterations, for a total of 1249 characters:

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+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-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-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+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+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+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-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

In each of these, the drawing tool starts at the first character and goes through until it reaches the end, following each instruction in order. So the first few characters in the above string would be read as:

“Draw a line. Turn right. Draw a line. Turn left. Draw a line. Turn left. Draw a line. Turn right. Draw a line. Turn right. Draw a line. Turn right.”

This can continue indefinitely. Complex starting conditions, combined with complex grammars and large numbers of iterations, can easily result in rule sets hundreds of millions of characters long.

After lines, angles are the most important in the creation of patterns in an L system. The following are a few examples of what differences in angles look like, based on the default state of the Explorer.

In all of these examples, clicking on the associated image will take you to the explorer, pre-configured to generate that image.

This is the initial condition, with an angle between lines of 90°:

Changing the angle to 60° results in this:

…And here is 45°:

Increasing past 90° makes the pattern more compact. Setting the angle to 120° creates a triangle pattern:

Increasing the iterations with the 120° pattern results in larger sets of nested triangles:

And finally, increasing the angle to 270° effective creates a mirror image of a pattern created with a 90° angle:

These should give you enough starting points to begin creating some interesting patterns. The angles which don’t precisely divide into 360 can result in some interesting interference patterns in the overlapping lines.