A Lindenmayer system (or L-system) is a parallel rewriting system consisting of an alphabet of symbols, a collection of rules that maps each symbol to another symbol, an initial starting string (the "axiom"), and a method to render an image from the result string (see the Wikipedia article for more information).
Within this project, the Lindenmayer class represents a generic L-system,
but it does not provide any facility to draw the result string. If each
symbol corresponds to an atomic action (i.e., there is no additional state
involved), then the SimpleLindenmayer class can be used which features a
mapping from symbols to functions used to draw the result string.
The pythagoras tree is an example of an L-system that requires additional state. It is constructed as follows:
- alphabet: 0, 1, [, ]
- axiom: 0
- rules: 1 → 11, 0 → 1[0]0
To draw the tree, the symbols are interpreted as follows:
- 0 and 1: draw a line segment
- [: push position and angle, turn left 45°
- ]: pop position and angle, turn right 45°
Here, push and pop refer to an internal stack used to navigate the tree.
The number of iterations controls how many branches are drawn; for example, with n = 5 the following tree results:
A square variant of the Koch curve can be constructed with the following L-system:
- alphabet: F, +, -
- axiom: F
- rules: F → F+F-F-F+F
The symbols are interpreted as follows:
- F: draw forward
- +: turn left 90°
- -: turn right 90°
For n = 4, the following image results:
A regular Koch snowflake can be constructed with the following L-system:
- alphabet: F, +, -
- axiom: F+F+F
- rules: F → F-F+F-F
As with the square Koch curve, F draws forward; + turns left by 60°, and - turns right by 120°. For n = 5, the following snowflake results:
The dragon curve can also be drawn with an L-system, as follows:
- alphabet: X, Y, F, +, -
- axiom: FX
- rules: X → X+YF+, Y → -FX-Y
As with the Koch curve, F draws forward and - and + turn left and right.
The Sierpinski triangle can also be approximated by an L-system.
- alphabet: A, B, +, -
- axiom: A
- rules: A → +B-A-B+, B → -A+B+A-
Here the symbols are interpreted like the Koch curve above, but A and B are used to mean 'draw forward'. For n = 8, the following triangle results:
L-systems can be used to draw realistic-looking plants. One such system is as follows:
- alphabet: X, F, +, - , [, ]
- axiom: X
- rules: X → F-[[X]+X]+F[+FX]-X, F → FF
Here as in the Pythagoras tree, [ and ] are used to push and pop the current position and angle onto an internal stack. The other symbols are:
- F: draw forward
- X: nothing (used to control the evolution of the curve)
- +: turn right 25°
- -: turn left 25°
For n = 6, the following plant results:
The Kochpinski is a majestic combination of a Koch curve and a Sierpinski triangle. For added majesty, make a Koch snowflake and fill in the middle with another Sierpinski triangle:
