algorithmfractalsl-systemsiterated-function

Is there an L-system representation for Polya's Triangle Sweep?

I am currently reading the book "The Fractal Geometry of Nature" by Benoit Mandelbrot. I tried to write a program to visualize some of the curves found in the book by using L-system replacement methods. For some of the curves one can find the rules quite easily, however for this curve I was not able to. Does anyone have a clue? I included a screenshot of the generator displayed in the book. Any help would be appreciated!

Generator of Polya's triangle sweep

Solution

  • According to this page, the equivalent L-system would be:

    Axiom: L
L = L+R-L-R
R = L+R+L-R

    For which we can write a short Python turtle program to confirm:

    from turtle import Screen, Turtle

AXIOM = "L"

RULES = {
    'L': "L+R-L-R",
    'R': "L+R+L-R"
}

LEVELS = 5
DISTANCE = 40

string = AXIOM

for _ in range(LEVELS):
    string = "".join(RULES.get(character, character) for character in string)

screen = Screen()

turtle = Turtle()
turtle.speed(0)  # because I have no patience

for character in string:
    if character in '-+':
        turtle.right([-90, 90][character == '+'])
    else:
        turtle.forward(DISTANCE / LEVELS)

screen.exitonclick()

    Pólya curve