If you start to zoom in on fractals, you will see the same shapes repeat themselves. Fractals are said to be self-similar, which means they can be drawn with recursion. This lab will walk you though drawing a Sierpinski triangle. Start by preparing the program to use Pythons turtle graphics. Sierpinski triangles can become quite complex, so increase the turtles speed to 10 (the maximum).

import turtle t = turtle.Turtle() t.speed(10) turtle.mainloop() 

The building block of this fractal is the triangle. Create a function (with a parameter for length) to draw a triangle. The turtle will be walking all over the screen, so it is important to make sure that the turtle is facing a consistent position before drawing the triangle. t.setheading(180) ensures the turtle is facing to the left.

import turtle t = turtle.Turtle() t.speed(10) def draw_triangle(length): t.setheading(180) for i in range(3): t.rt(120) t.fd(length) draw_triangle(50) turtle.mainloop() 

TRY IT

Look closely at a Sierpinski triangle, and you will see clusters of three triangles that make up clusters of triangles and so forth.

You are now going to create a recursive function that draws this cluster of three triangles. Define the function sierpinski that takes length and n as parameters. The base case is if n is equal to 1. If so, draw a triangle of size length. If n is not equal to 1, then you are going to call sierpinski again, but with n-1. These new triangles need to be in a different position, so move the turtle after drawing each turtle. Finally, replace the draw_triangle function call with sierpinski(50, 1).

import turtle t = turtle.Turtle() t.speed(10) def sierpinski(length, n): if n == 1: draw_triangle(length) else: sierpinski(length, n-1) t.rt(120) t.fd(length) sierpinski(length, n-1) t.lt(120) t.fd(length) sierpinski(length, n-1) t.fd(length) def draw_triangle(length): t.setheading(180) for i in range(3): t.rt(120) t.fd(length) sierpinski(50, 1) turtle.mainloop()

What happens if you:

Change the function call to sierpinski(50, 2)?

Change the function call to sierpinski(50, 3)?

Change the function call to sierpinski(50, 4)?

TRY IT

The triangles are clustered together, but the Sierpinski triangle has larger triangle-shaped voids. An adjustment needs to be made to the distance the turtle moves between calls to the sierpinski function. Instead of moving forward the distance of length, the turtle will move forward length * (n-1). Change the sierpinski function call to sierpinski(20, 4).

import turtle t = turtle.Turtle() t.speed(10) def sierpinski(length, n): if n == 1: draw_triangle(length) else: sierpinski(length, n-1) t.rt(120) t.fd(length * (n-1)) sierpinski(length, n-1) t.lt(120) t.fd(length * (n-1)) sierpinski(length, n-1) t.fd(length * (n-1)) def draw_triangle(length): t.setheading(180) for i in range(3): t.rt(120) t.fd(length) sierpinski(20, 4) turtle.mainloop() 

TRY IT

The fractal is getting better, but there are a few areas where the program can be improved. Change the distance the turtle goes forward from t.fd(length * (n-1)) to t.fd(length * 2 ** (n-2)).

import turtle t = turtle.Turtle() t.speed(10) def sierpinski(length, n): if n == 1: draw_triangle(length) else: sierpinski(length, n-1) t.rt(120) t.fd(length * 2**(n-2)) sierpinski(length, n-1) t.lt(120) t.fd(length * 2**(n-2)) sierpinski(length, n-1) t.fd(length * 2**(n-2)) def draw_triangle(length): t.setheading(180) for i in range(3): t.rt(120) t.fd(length) sierpinski(20, 4) turtle.mainloop() 

TRY IT

What happens if you:

Change the sierpinski function call to sierpinski(5, 6)?

Change the sierpinski function call to sierpinski(5, 8)?

TRY IT

Lab Question

What is the relationship between the draw_triangle function and the sierpinski function?

sierpinski is a helper function for draw_triangle

draw_triangle is a helper function for sierpinski

draw_triangle is declared inside sierpinski

draw_triangle is called as a parameter for the sierpinski function