3. Exercises¶

1. Here is another interesting L-System called a Hilbert curve. Use 90 degrees:

   L
   L -> +RF-LFL-FR+
   R -> -LF+RFR+FL-
   

   import turtle def createLSystem(numIters, axiom): startString = axiom endString = "" for i in range(numIters): endString = processString(startString) startString = endString return endString def processString(oldStr): newstr = "" for ch in oldStr: newstr = newstr + applyRules(ch) return newstr def applyRules(ch): newstr = "" if ch == 'L': newstr = '+RF-LFL-FR+' # Rule 1 elif ch == 'R': newstr = '-LF+RFR+FL-' else: newstr = ch # no rules apply so keep the character return newstr def drawLsystem(aTurtle, instructions, angle, distance): for cmd in instructions: if cmd == 'F': aTurtle.forward(distance) elif cmd == 'B': aTurtle.backward(distance) elif cmd == '+': aTurtle.right(angle) elif cmd == '-': aTurtle.left(angle) def main(): inst = createLSystem(4, "L") # create the string print(inst) t = turtle.Turtle() # create the turtle wn = turtle.Screen() t.up() t.back(200) t.down() t.speed(9) drawLsystem(t, inst, 90, 5) # draw the picture # angle 90, segment length 5 wn.exitonclick() main()

2. Here is a dragon curve. Use 90 degrees.:

   FX
   X -> X+YF+
   Y -> -FX-Y
   

3. Here is something called an arrowhead curve. Use 60 degrees.:

   YF
   X -> YF+XF+Y
   Y -> XF-YF-X
   

   import turtle def createLSystem(numIters, axiom): startString = axiom endString = "" for i in range(numIters): endString = processString(startString) startString = endString return endString def processString(oldStr): newstr = "" for ch in oldStr: newstr = newstr + applyRules(ch) return newstr def applyRules(ch): newstr = "" if ch == 'X': newstr = 'YF+XF+Y' # Rule 1 elif ch == 'Y': newstr = 'XF-YF-X' else: newstr = ch # no rules apply so keep the character return newstr def drawLsystem(aTurtle, instructions, angle, distance): for cmd in instructions: if cmd == 'F': aTurtle.forward(distance) elif cmd == 'B': aTurtle.backward(distance) elif cmd == '+': aTurtle.right(angle) elif cmd == '-': aTurtle.left(angle) def main(): inst = createLSystem(5, "YF") # create the string print(inst) t = turtle.Turtle() # create the turtle wn = turtle.Screen() t.speed(9) drawLsystem(t, inst, 60, 5) # draw the picture # angle 90, segment length 5 wn.exitonclick() main()

4. Try the Peano-Gosper curve. Use 60 degrees.:

   FX
   X -> X+YF++YF-FX--FXFX-YF+
   Y -> -FX+YFYF++YF+FX--FX-Y
   

5. The Sierpinski Triangle. Use 60 degrees.:

   FXF--FF--FF
   F -> FF
   X -> --FXF++FXF++FXF--
   

   import turtle def createLSystem(numIters, axiom): startString = axiom endString = "" for i in range(numIters): endString = processString(startString) startString = endString return endString def processString(oldStr): newstr = "" for ch in oldStr: newstr = newstr + applyRules(ch) return newstr def applyRules(ch): newstr = "" if ch == 'F': newstr = 'FF' # Rule 1 elif ch == 'X': newstr = '--FXF++FXF++FXF--' else: newstr = ch # no rules apply so keep the character return newstr def drawLsystem(aTurtle, instructions, angle, distance): for cmd in instructions: if cmd == 'F': aTurtle.forward(distance) elif cmd == 'B': aTurtle.backward(distance) elif cmd == '+': aTurtle.right(angle) elif cmd == '-': aTurtle.left(angle) def main(): inst = createLSystem(5, "FXF--FF--FF") # create the string print(inst) t = turtle.Turtle() # create the turtle wn = turtle.Screen() t.up() t.back(200) t.right(90) t.forward(100) t.left(90) t.down() t.speed(9) drawLsystem(t, inst, 60, 5) # draw the picture # angle 90, segment length 5 wn.exitonclick() main()

6. Here are the rules for an L-system that creates something that resembles a common garden herb. Implement the following rules and try it. Use an angle of 25.7 degrees.

   H
   H --> HFX[+H][-H]
   X --> X[-FFF][+FFF]FX
   

   import turtle def createLSystem(numIters, axiom): startString = axiom endString = "" for i in range(numIters): endString = processString(startString) startString = endString return endString def processString(oldStr): newstr = "" for ch in oldStr: newstr = newstr + applyRules(ch) return newstr def applyRules(ch): newstr = "" if ch == 'H': newstr = 'HFX[+H][-H]' # Rule 1 elif ch == 'X': newstr = 'X[-FFF][+FFF]FX' else: newstr = ch # no rules apply so keep the character return newstr def drawLsystem(aTurtle, instructions, angle, distance): savedInfoList = [] for cmd in instructions: if cmd == 'F': aTurtle.forward(distance) elif cmd == 'B': aTurtle.backward(distance) elif cmd == '+': aTurtle.right(angle) elif cmd == '-': aTurtle.left(angle) elif cmd == '[': savedInfoList.append([aTurtle.heading(), aTurtle.xcor(), aTurtle.ycor()]) #print(savedInfoList) elif cmd == ']': newInfo = savedInfoList.pop() aTurtle.setheading(newInfo[0]) aTurtle.setposition(newInfo[1], newInfo[2]) def main(): inst = createLSystem(4, "H") # create the string print(inst) t = turtle.Turtle() # create the turtle wn = turtle.Screen() t.up() t.back(200) t.down() t.speed(9) drawLsystem(t, inst, 27.5, 5) # draw the picture wn.exitonclick() main()

7. Here is another L-System. Use an Angle of 25.

   F
   F --> F[-F]F[+F]F