L-systems list


What is L-system?

It is fractal polyline. It is defined by an axiom (a string), rules of interplacing and an rotation angle.

First, we need to take the axiom and make interplacings according to rules: for rule 'x > y' we replace all 'x' substrings of axiom with 'y'. We do it for several iterations and obtain a string, whch is a program for 'turtle'. The turtle crawls on the plane and leaves a trail. It can do following commands (they aren't common, but in the list below these commands are used):

All other symbols are ignored.

More information on Wikipedia.

L-systems online.


Koch's curveKoch's curve
Axiom: F
Rule: F > F-F++F-F
Angle: 60
Koch's curve
Koch's snowflakeKoch's snowflake
Axiom: F++F++F
Rule: F > F-F++F-F
Angle: 60
Koch's snowflake
Axiom: F+F+F
Rule: F > F-F+F
Angle: 120
Axiom: F+F+F+F
Rule: F > FF+F++F+F
Angle: 90
Ice fractals
Кривая драконаDragon curve
Axiom: FX
Rules:
X > X+YF+
Y > -FX-Y
Angle: 90
Dragon curve
Gosper's curve
Axiom: XF
Rules:
X > X+YF++YF-FX--FXFX-YF+
Y > -FX+YFYF++YF+FX--FX-Y
Angle: 60
Gosper's curve
Serpinski curve
Axiom: F+XF+F+XF
Rule:
X > XF-F+F-XF+F+XF-F+F-X
Angle: 90
Gilbert's curve
Axiom: X
Rules:
X > -YF+XFX+FY-
Y > +XF-YFY-FX+
Angle: 90
Gilbert's curve
Axiom: F+F+F+F
Rule: F > FF+F+F+F+FF
Angle: 90
Axiom: F+F+F+F
Rule: F > F+F-F-FF+F+F-F
Angle: 90
Generalizations of the Koch's curve<>
Axiom: F+F+F+F
Rule: F > F+F-F-FFF+F+F-F
Angle: 90
Generalizations of the Koch's curve
Axiom: F+F+F+F
Rule: F > F-FF+FF+F+F-F-FF+F+F-F-FF-FF+F
Angle: 90
Generalizations of the Koch's curve
Quadratic Snowflake
Axiom: F
Rule: F > F-F+F+F-F
Angle: 90
Axiom: YF
Rules:
X > YF+XF+Y
Y > XF-YF-X
Angle: 60
Axiom: F+F+F+F
Rule: F > F+F-F+F+F
Angle: 90
Axiom: F+F+F+F
Rule: F > FF+F+F+F+F+F-F
Angle: 90
A bush
Axiom: Y
Rules:
X > X[-FFF][+FFF]FX
Y > YFX[+Y][-Y]
Angle: π/7
A bush
Axiom: F
Rule: F > FF+[+F-F-F]-[-F+F+F]
Angle: π/8
P. Bourke L-System Bush
A bush
Axiom: F
Rule: F > F[+FF][-FF]F[-F][+F]F
Angle: π/5
A bush
Axiom: X
Rules:
F > FF
X > F[+X]F[-X]+X
Angle: π/9
A bush
Axiom: F-F-F-F
Rule: F > F-F+F+F-F
Angle: 90
MathWorld Box Fractal
Сорняк
Axiom: F
Rule: F > F[+F]F[-F]F
Angle: π/7
Из галереи Дмитрия Зайкова
Axiom: F
Rules:
F > FXF
X > [-F+F+F]+F-F-F+
Angle: 60
Из галереи Дмитрия Зайкова
Serpinski triangle
Axiom: FXF--FF--FF
Rules:
F > FF
X > --FXF++FXF++FXF--
Angle: 60
Serpinski triangle
Serpinski carpet
Axiom: F
Rule: F > FFF[+FFF+FFF+FFF]
Angle: 90°
Serpinski carpet
Мозаика
Axiom: F-F-F-F
Rules:
F > F-b+FF-F-FF-Fb-FF+b-FF+F+FF+Fb+FFF
b > bbbbbb
Angle: 90°
Здесь b означает переместиться вперёд на один шаг, не прорисовывая след.
Р. М. Кроновер, Фракталы и хаос динамических системах, стр. 36.
Levy curve
Axiom: F++F++F++F
Rule: F > -F++F-
Angle: 45°
Кривая Леви
Lévy C curve
Axiom: F++F++F++F++F
Rule: F > F++F++F+++++F-F++F
Angle: 36°
P. Bourke Pentaplexity
Axiom: F
Rule: F > F+F-F
Angle: 120°
William McWorter Terdragon
Sierπnski Carpet
Axiom: F
Rules:
F > F+F-F-F-b+F+F+F-F
b > bbb
Angle: 90°
Здесь b означает переместиться вперёд на один шаг, не прорисовывая след.
William McWorter Sierπnski Carpet
Ковёр Серпинского
Axiom: F-F-F-F-F
Rule: F > F-F++F+F-F-F
Angle: 72°
William McWorter Pentigree
Axiom: X
Rules:
F >
X > -F++F-X-F--F+Y---F--F+Y+F++F-X+++F++F-X-F++F-X+++F--F+Y--
Y > +F++F-X-F--F+Y+F--F+Y---F--F+Y---F++F-X+++F++F-X+++F--F+Y
Angle: 30°
Gary Teachout Hex-7-b
Axiom: FX
Rules:
F >
X > FX-FY-FX+FY+FX+FY+FX+FY+FX-FY-FX-FY-FX-FY-FX+FY+FX
Y > FY
Angle: 45°
Gary Teachout Peano-c
Axiom: XYXYXYX+?XYXYXYX+?XYXYXYX+?XYXYXYX
Rules:
F >
X > FX+FX+FXFY-FY-
Y > +FX+FXFY-FY-FY
Angle: 90°
William McWorter Border1
Axiom: F--F--F--F--F--F
Rule: F > -F[--F--F]++F--F+
Angle: 30°
Adrian Mariano Doily
Axiom: F+F+F
Rule: F > F+FF-F
Angle: 120°
William McWorter Maze01
Axiom: X
Rules:
F >
X > FY+FYFY-FY
Y > FX-FXFX+FX
Angle: 120°
William McWorter Maze&Fractal1
Axiom: X
Rules:
F >
X > FX+FX+FXFYFX+FXFY-FY-FY-
Y > +FX+FX+FXFY-FYFXFY-FY-FY
Angle: 90°
William McWorter Moore
Axiom: X-X-X-X-X
Rules:
F >
X > FX-FX-FX+FY+FY+FX-FX
Y > FY+FY-FX-FX-FY+FY+FY
Angle: 72°
William McWorter Pentant
Axiom: F-F-F-F-F
Rule: F > F-F-F++F+F-F
Angle: 72°
William McWorter Pentl
Axiom: L--F--L--F
Rules:
L > +R-F-R+
R > -L+F+L-
Angle: 45°
William McWorter Sierπnsk
Axiom: X
Rules:
X > F-F-F+F+FX++F-F-F+F+FX--F-F-F+F+FX
F >
Angle: 60°
William McWorter Tiling1
Axiom: F++++F
Rule: F > F+F+F++++F+F+F
Angle: 45°
Anthony Hanmer ADH231a
Axiom: F+F+F+F++F-F-F-F
Rule: F > F+F++F+FF
Angle: 90°
Anthony Hanmer ADH256a
Axiom: F++F++F+++F--F--F
Rule: F > FF++F++F++FFF
Angle: 60°
Anthony Hanmer ADH258a


Materials from the site fractalworld.xaoc.ru were used.