L-systems: Difference between revisions

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===Lindenmayer Systems===
===Online experiments with Lindenmayer Systems===
A Lindenmayer System consists of
A Lindenmayer System consists of
* an initial string called ''axiom''
* an initial string called ''axiom''
* a set of rewriting ''rules''
* a set of rewriting ''rules''


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


===Examples===
Most of the examples are from the book by Przemyslaw Prusinkiewicz and James Hanan: ''Lindenmayer Systems, Fractals, and Plants'', see the References.
Most of the examples are from the book by Przemyslaw Prusinkiewicz and James Hanan: ''Lindenmayer Systems, Fractals, and Plants'', see the References.
 
====First Example: Sierpinski curve====
===Sierpinski curve===
<html>
<html>
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxturtle.js"></script>
<form><textarea id="inputtext0" rows=15 cols=35 wrap="off" style="width:600px;">
<form><textarea id="inputtext0" rows=3 cols=35 wrap="off" style="width:600px;">
var level = 6;
var level = 6;
var axiom = 'A';
var axiom = 'A';
Line 39: Line 37:
<input type="button" value="clear" onClick="clearturtle(0)">
<input type="button" value="clear" onClick="clearturtle(0)">
</form>
</form>
<div id="box0" class="jxgbox" style="width:600px; height:600px;"></div>
</html>
<script language="JavaScript">
<jsxgraph width="600" height="600" box="box0">
var t;
var t;
var turtle = [];
var turtle = [];
var brd = [];
var brd = [];
brd[0] = JXG.JSXGraph.initBoard('box0', {originX: 300, originY: 300, unitX: 1, unitY: 1});
brd[0] = JXG.JSXGraph.initBoard('box0', {boundingbox: [-300, 300, 300, -300]});
turtle[0] = new JSXTurtleObj(brd[0]);
turtle[0] = brd[0].create('turtle');


function expander(level,axiom,rules) {
function expander(level,axiom,rules) {
Line 56: Line 54:
         this.next = function() {
         this.next = function() {
             if (this.pos>=this.code.length) return null;
             if (this.pos>=this.code.length) return null;
             return this.code[this.pos++];
             return this.code.charAt(this.pos++);
         }
         }
     });
     });
Line 69: Line 67:
             this.code = this.rules[pattern];
             this.code = this.rules[pattern];
         }
         }
         return this.code[this.pos++];
         return this.code.charAt(this.pos++);
     }
     }
}
}
Line 108: Line 106:
function run(nr) {
function run(nr) {
   brd[nr].suspendUpdate();
   brd[nr].suspendUpdate();
   var code = $('inputtext'+nr).value;
   var code = document.getElementById('inputtext'+nr).value;
   if (code=='') { return; }
   if (code=='') { return; }
   t = turtle[nr];
   t = turtle[nr];
Line 122: Line 120:
   turtle[nr].cs();
   turtle[nr].cs();
}
}
</script>
</jsxgraph>
</html>
 
 
===Quadratic snowflake variation===
 
<html>
<form><textarea id="inputtext1" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(1)">
<input type="button" value="clear" onClick="clearturtle(1)">
</form>
<div id="box1" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[1] = JXG.JSXGraph.initBoard('box1', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[1] = new JSXTurtleObj(brd[1]);
</script>
</html>
 
===Dragon curve===
 
<html>
<form><textarea id="inputtext2" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(2)">
<input type="button" value="clear" onClick="clearturtle(2)">
</form>
<div id="box2" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[2] = JXG.JSXGraph.initBoard('box2', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[2] = new JSXTurtleObj(brd[2]);
</script>
</html>
 
===Islands and lakes===
 
<html>
<form><textarea id="inputtext3" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(3)">
<input type="button" value="clear" onClick="clearturtle(3)">
</form>
<div id="box3" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[3] = JXG.JSXGraph.initBoard('box3', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[3] = new JSXTurtleObj(brd[3]);
</script>
</html>
 
 
===Peano curve===
 
<html>
<form><textarea id="inputtext4" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(4)">
<input type="button" value="clear" onClick="clearturtle(4)">
</form>
<div id="box4" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[4] = JXG.JSXGraph.initBoard('box4', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[4] = new JSXTurtleObj(brd[4]);
</script>
</html>


===Hexagonal Gosper curve===
====More examples====
* [[Quadratic snowflake variation]]
* [[Dragon curve]]
* [[Islands and lakes]]
* [[Peano curve]]
* [[Hexagonal Gosper curve]]
* [[Plant generation I]]
* [[Plant generation II]]
* [[Hexagonal kolam]]
* [[Mango kolam]]
* [[Penrose tiling]]


<html>
===The underlying JavaScript code===
<form><textarea id="inputtext5" rows=3 cols=35 wrap="off" style="width:600px;">
The underlying [[JavaScript code for producing Lindenmayer systems]]
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);
</textarea><br />
<input type="button" value="run" onClick="run(5)">
<input type="button" value="clear" onClick="clearturtle(5)">
</form>
<div id="box5" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[5] = JXG.JSXGraph.initBoard('box5', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[5] = new JSXTurtleObj(brd[5]);
</script>
</html>
 
===Plant 1===
 
<html>
<form><textarea id="inputtext6" rows=3 cols=35 wrap="off" style="width:600px;">
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});
</textarea><br />
<input type="button" value="run" onClick="run(6)">
<input type="button" value="clear" onClick="clearturtle(6)">
</form>
<div id="box6" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[6] = JXG.JSXGraph.initBoard('box6', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[6] = new JSXTurtleObj(brd[6]);
</script>
</html>
 
===Plant 2===
 
<html>
<form><textarea id="inputtext7" rows=3 cols=35 wrap="off" style="width:600px;">
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});
</textarea><br />
<input type="button" value="run" onClick="run(7)">
<input type="button" value="clear" onClick="clearturtle(7)">
</form>
<div id="box7" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[7] = JXG.JSXGraph.initBoard('box7', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[7] = new JSXTurtleObj(brd[7]);
</script>
</html>
 
===Hexagonal kolam===
 
<html>
<form><textarea id="inputtext8" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(8)">
<input type="button" value="clear" onClick="clearturtle(8)">
</form>
<div id="box8" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[8] = JXG.JSXGraph.initBoard('box8', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[8] = new JSXTurtleObj(brd[8]);
</script>
</html>
 
===Mango kolam===
 
<html>
<form><textarea id="inputtext9" rows=3 cols=35 wrap="off" style="width:600px;">
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);
</textarea><br />
<input type="button" value="run" onClick="run(9)">
<input type="button" value="clear" onClick="clearturtle(9)">
</form>
<div id="box9" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[9] = JXG.JSXGraph.initBoard('box9', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[9] = new JSXTurtleObj(brd[9]);
</script>
</html>
 
 
===Penrose tiling===
 
<html>
<form><textarea id="inputtext10" rows=3 cols=35 wrap="off" style="width:600px;">
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'});
</textarea><br />
<input type="button" value="run" onClick="run(10)">
<input type="button" value="clear" onClick="clearturtle(10)">
</form>
<div id="box10" class="jxgbox" style="width:600px; height:600px;"></div>
<script language="JavaScript">
brd[10] = JXG.JSXGraph.initBoard('box10', {originX: 300, originY: 300, unitX: 1, unitY: 1});
turtle[10] = new JSXTurtleObj(brd[10]);
</script>
</html>


===References===
===References===
Line 495: Line 141:
* [http://en.wikipedia.org/wiki/L-system http://en.wikipedia.org/wiki/L-system]
* [http://en.wikipedia.org/wiki/L-system http://en.wikipedia.org/wiki/L-system]
* [http://en.wikipedia.org/wiki/Space-filling_curve http://en.wikipedia.org/wiki/Space-filling_curve]
* [http://en.wikipedia.org/wiki/Space-filling_curve http://en.wikipedia.org/wiki/Space-filling_curve]
* [http://en.wikipedia.org/wiki/Penrose_tiling http://en.wikipedia.org/wiki/Penrose_tiling]
* [http://www.biologie.uni-hamburg.de/b-online/e28_3/lsys.html An Introduction to Lindenmayer Systems]
* [http://en.wikipedia.org/wiki/Kolam http://en.wikipedia.org/wiki/Kolam]




Line 502: Line 147:
[[Category:Fractals]]
[[Category:Fractals]]
[[Category:Turtle Graphics]]
[[Category:Turtle Graphics]]
[[Category:Lindenmayer systems]]

Latest revision as of 07:59, 8 June 2011

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.

Examples

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

First Example: Sierpinski curve


More examples

The underlying JavaScript code

The underlying JavaScript code for producing Lindenmayer systems

References

Retrieved from "http://jsxgraph.org/wiki/index.php?title=L-systems&oldid=5223"