L-systems

Online experiments with Lindenmayer Systems

A Lindenmayer System consists of

• an initial string called axiom
• a set of rewriting rules

This is an experimental page, where the Lindenmayer Systems can be changed online. The visualization is done by the JavaScript library JSXGraph. For each system a maximum level is defined. If this value is increased, the complexity of the drawing rises and running time increases, too.

Most of the examples are from the book by Przemyslaw Prusinkiewicz and James Hanan: Lindenmayer Systems, Fractals, and Plants, see the References.

Sierpinski curve

var level = 6; var axiom = 'A'; var rules = { 'A':'B-A-B', 'B':'A+B+A', '+' : '+', '-' : '-' }; var symbols = { 'A':'F', 'B':'F', '+':'+', '-':'-', '[':'[', ']':']' }; var angle = 60; var len = 500/Math.pow(2,level); t.setPos(-250*Math.pow(-1,level),-250); t.rt(90*Math.pow(-1,level));

• [Quadratic snowflake variation]

Quadratic snowflake variation

var level = 5; var axiom = '+F'; var rules = { 'F':'F-F+F+F-F', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 90; var len = 500/Math.pow(3,level); t.setPos(250,0);

Dragon curve

var level = 8; var axiom = 'Fl'; var rules = { 'F' : 'F', 'l' : 'l+rF+', 'r' : '-Fl-r', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'l':' ', 'r':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 90; var len = 500/(level*level);

Islands and lakes

var level = 2; var axiom = 'F-F-F-F'; var rules = { 'F' : 'F-f+FF-F-FF-Ff-FF+f-FF+F+FF+Ff+FFF', 'f' : 'ffffff', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'f':'f', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 90; var len = 20/Math.pow(2,level);

Peano curve

var level = 4; var axiom = 'X'; var rules = { 'F' : 'F', 'X' : 'XFYFX+F+YFXFY-F-XFYFX', 'Y' : 'YFXFY-F-XFYFX+F+YFXFY', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'X':' ', 'Y':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 90; var len = 500/Math.pow(3,level); t.setPos(250,-250);

Hexagonal Gosper curve

var level = 3; var axiom = 'XF'; var rules = { 'F' : 'F', 'X' : 'X+YF++YF-FX--FXFX-YF+', 'Y' : '-FX+YFYF++YF+FX--FX-Y', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'X':' ', 'Y':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 60; var len = 500/Math.pow(3,level); t.setPos(250,0);

Plant 1

var level = 3; var axiom = 'F'; var rules = { 'F' : 'F[+F]F[-F]F', '[' : '[', ']' : ']', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 25.7; var len = 500/Math.pow(3,level); t.setPos(0,-250); var shrink = 1.0; t.setProperty({strokeColor:'green',strokeWidth:2});

Plant 2

var level = 4; var axiom = 'X'; var rules = { 'F' : 'FF', 'X' : 'F-[[X]+X]+F[+FX]-X', '[' : '[', ']' : ']', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'X':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 22.5; var len = 800/Math.pow(3,level); t.setPos(0,-250); t.setProperty({strokeColor:'green',strokeWidth:2});

Hexagonal kolam

var level = 4; var axiom = 'X'; var rules = { 'F' : 'F', 'X' : '[-F+F[Y]+F][+F-F[X]-F]', 'Y' : '[-F+F[Y]+F][+F-F-F]', '[' : '[', ']' : ']', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'X':' ', 'Y':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 60; var len = 300/(level); t.setPos(0,-250);

Mango kolam

var level = 5; var axiom = 'A---A'; var rules = { 'F' : 'F', 'f' : 'f', 'A' : 'f-F+Z+F-fA', 'Z' : 'F-FF-F--[--Z]F-FF-F--F-FF-F--', '[' : '[', ']' : ']', '+' : '+', '-' : '-' }; var symbols = { 'F':'F', 'f':'f', 'A':' ', 'Z':' ', '+':'+', '-':'-', '[':'[', ']':']' } ; var angle = 60; var len = 100/(level);

Penrose tiling

var level = 2; var axiom = '[7]++[7]++[7]++[7]++[7]'; var rules = { '6' : '81++91----71[-81----61]++', '7' : '+81--91[---61--71]+', '8' : '-61++71[+++81++91]-', '9' : '--81++++61[+91++++71]--71', '1' : '', '+' : '+', '-' : '-', '[' : '[', ']' : ']' }; var symbols = { '1':'F', '6':' ', '7':' ', '8':' ', '9':' ', '+':'+', '-':'-', '[':'[', ']':']' }; var angle = 36.0; var len = 100/(level); t.setProperty({fillcolor:'#abff00'});

The underlying JavaScript code

The underlying JavaScript code for producing Lindenmayer systems

References

• Przemyslaw Prusinkiewicz, James Hanan: Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics). Springer-Verlag 1989, ISBN 0-387-97092-4
• http://en.wikipedia.org/wiki/L-system
• http://en.wikipedia.org/wiki/Space-filling_curve
• http://en.wikipedia.org/wiki/Penrose_tiling
• http://en.wikipedia.org/wiki/Kolam