This is a Python implementation of L-systems.
The following is a gallery of various mathematical objects that can be described by an L-system.
- Algae
- Thue-Morse sequence
- Fractal binary tree
- Cantor set
- Koch curves
- Sierpiński triangles
- Sierpiński carpet
- Sierpiński curves
- Dragon curves
- Fractal plant
- Hilbert curve
- Peano curve
- Gosper curves
- Penrose tiling (P3)
- Pentigrees
- variables:
A,B - constants: none
- axiom:
A - rules:
A→AB,B→A
n = 0: A
n = 1: AB
n = 2: ABA
n = 3: ABAAB
n = 4: ABAABABA
n = 5: ABAABABAABAAB
n = 6: ABAABABAABAABABAABABA
n = 7: ABAABABAABAABABAABABAABAABABAABAAB
- variables:
0,1 - constants: none
- axiom:
0 - rules:
0→01,1→10
n = 0: 0
n = 1: 01
n = 2: 0110
n = 3: 01101001
n = 4: 0110100110010110
n = 5: 01101001100101101001011001101001
n = 6: 0110100110010110100101100110100110010110011010010110100110010110
- variables:
0,1 - constants:
[,] - axiom:
0 - rules:
1→11,0→1[0]0
0: go forward with drawing a green line segment1: go forward with drawing a brown line segment[: push the current pose on the stack, turn 45° to the left[: pop a pose from the stack, turn 45° to the right
- variables:
F,f - constants: none
- axiom:
F - rules:
F→FfF,f→fff
F: go forward with drawing a line segmentf: go forward without drawing
- variables:
F - constants:
+,- - axiom:
F--F--F - rules:
F→F+F--F+F
F: go forward with drawing a line segment+: turn 60° to the left-: turn 60° to the right
- variables:
F - constants:
+,- - axiom:
F - rules:
F→F+F-F-F+F
F: go forward with drawing a line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F - constants:
+,- - axiom:
F - rules:
F→F+F-F-FF+F+F-F
F: go forward with drawing a line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F,G - constants:
+,- - axiom:
F-G-G - rules:
F→F-G+F+G-F,G→GG
F: go forward with drawing a red line segmentG: go forward with drawing a yellow line segment+: turn 120° to the left-: turn 120° to the right
- variables:
F,G - constants:
+,- - axiom:
F - rules:
F→G-F-G,G→F+G+F
F: go forward with drawing a red line segmentG: go forward with drawing a blue line segment+: turn 60° to the left-: turn 60° to the right
- variables:
F,f,X - constants:
+,- - axiom:
F+F+F+F+X - rules:
F→FFF,f→fff,X→XfXfX+fF++ff-f-fF++ff-f-f-XfFX++ff-f-XfXFX++ff+ff+
F: go forward with drawing a line segmentf: go forward without drawingX: point a blue dot+: turn 120° to the left-: turn 120° to the right
- variables:
X - constants:
F,+,- - axiom:
F--XF--F--XF - rules:
X→XF+G+XF--F--XF+G+X
X: do nothingF: go forward with drawing an orange line segmentG: go forward with drawing a brown line segment+: turn 45° to the left-: turn 45° to the right
- variables:
X - constants:
F,+,- - axiom:
F+XF+F+XF - rules:
X→XF-F+F-XF+F+XF-F+F-X
X: do nothingF: go forward with drawing a line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F,G - constants:
+,- - axiom:
F - rules:
F→F+G,G→F-G
F: go forward with drawing a red line segmentG: go forward with drawing a green line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F,G,X,Y - constants:
+,- - axiom:
F+G+X+Y - rules:
F→F+G,G→F-G,X→X+Y,Y→X-Y
F: go forward with drawing a red line segmentG: go forward with drawing a red line segmentX: go forward with drawing a blue line segmentY: go forward with drawing a blue line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F,G,H - constants:
+,- - axiom:
F+G-H - rules:
F→F+F-F,G→G+G-G,H→H+H-H
F: go forward with drawing a red line segmentG: go forward with drawing a green line segmentH: go forward with drawing a blue line segment+: turn 120° to the left-: turn 120° to the right
- variables:
F - constants:
+,- - axiom:
F - rules:
F→+F--F+
F: go forward with drawing a line segment+: turn 45° to the left-: turn 45° to the right
- variables:
F,X - constants:
+,-,[,] - axiom:
X - rules:
F→FF,X→F+[[X]-X]-F[-FX]+X
F: go forward with drawing a line segmentX: do nothing+: turn 25° to the left-: turn 25° to the right[: push the current pose on the stack]: pop a pose from the stack
- variables:
X,Y - constants:
F,+,- - axiom:
X - rules:
X→+YF-XFX-FY+,Y→-XF+YFY+FX-
F: go forward with drawing a line segmentX: do nothingY: do nothing+: turn 90° to the left-: turn 90° to the right
- variables:
X,Y - constants:
F,+,- - axiom:
X - rules:
X→XFYFX-F-YFXFY+F+XFYFX,Y→YFXFY+F+XFYFX-F-YFXFY
F: go forward with drawing a line segmentX: do nothingY: do nothing+: turn 90° to the left-: turn 90° to the right
- variables:
F,G - constants:
+,- - axiom:
F - rules:
F→F-G--G+F++FF+G-,G→+F-GG--G-F++F+G
F: go forward with drawing a green line segmentG: go forward with drawing a blue line segment+: turn 60° to the left-: turn 60° to the right
- variables:
F,G - constants:
+,- - axiom:
G - rules:
F→FF-G-G+F+F-G-GF+G+FFG-F+G+FF+G-FG-G-F+F+GG-,G→+FF-G-G+F+FG+F-GG-F-G+FGG-F-GF+F+G-G-F+F+GG
F: go forward with drawing a green line segmentG: go forward with drawing a blue line segment+: turn 90° to the left-: turn 90° to the right
- variables:
F,1,2,3,4 - constants:
+,-,[,] - axiom:
[2]++[2]++[2]++[2]++[2] - rules:
F→(empty)1→3F++4F----2F[-3F----1F]++2→+3F--4F[---1F--2F]+3→-1F++2F[+++3F++4F]-4→--3F++++1F[+4F++++2F]--2F
F: go forward with drawing a line segment1: do nothing2: do nothing3: do nothing4: do nothing+: turn 36° to the left-: turn 36° to the right[: push the current pose on the stack]: pop a pose from the stack
- variables:
1,2,3,4,5 - constants:
+,- - axiom:
1++2++3++4++5 - rules:
1→+1++1----1--1++1++1-2→+2++2----2--2++2++2-3→+3++3----3--3++3++3-4→+4++4----4--4++4++4-5→+5++5----5--5++5++5-
1: go forward with drawing a red line segment2: go forward with drawing a yellow line segment3: go forward with drawing a green line segment4: go forward with drawing a blue line segment5: go forward with drawing a purple line segment+: turn 36° to the left-: turn 36° to the right
- variables:
1,2,3,4,5 - constants:
+,- - axiom:
1+2+3+4+5 - rules:
1→1+1-1--1+1+12→2+2-2--2+2+23→3+3-3--3+3+34→4+4-4--4+4+45→5+5-5--5+5+5
1: go forward with drawing a red line segment2: go forward with drawing a yellow line segment3: go forward with drawing a green line segment4: go forward with drawing a blue line segment5: go forward with drawing a purple line segment+: turn 72° to the left-: turn 72° to the right
- Przemyslaw Prusinkiewicz and Aristid Lindenmayer. The Algorithmic Beauty of Plants, Springer-Verlag, 1990.
- Lawrence H. Riddle, Classic Iterated Function Systems, https://larryriddle.agnesscott.org/ifs/ifs.htm
- StackExchange - Does there exist an L-system for the sierpiński curve, https://math.stackexchange.com/questions/3393187
- Wikipedia articles
- https://en.wikipedia.org/wiki/L-system
- https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence
- https://en.wikipedia.org/wiki/Koch_snowflake
- https://en.wikipedia.org/wiki/Sierpi%C5%84ski_curve
- https://en.wikipedia.org/wiki/Hilbert_curve
- https://en.wikipedia.org/wiki/Peano_curve
- https://en.wikipedia.org/wiki/Gosper_curve
- https://en.wikipedia.org/wiki/Dragon_curve
- https://es.wikipedia.org/wiki/Teselaci%C3%B3n_de_Penrose