Koch Snowflake in Python Turtle

Intermediate · Fractals · 48 lines · draws in about 13 seconds

Each side of a triangle becomes a Koch line, and a Koch line is four smaller Koch lines with a bump in the middle, four levels deep.

Koch Snowflake in Python Turtle: the finished drawing

import turtle

import math

screen = turtle.Screen()

screen.bgcolor("midnightblue")

screen.title("Koch snowflake")

t = turtle.Turtle()

t.speed(10)

t.pensize(2)

t.color("white", "lightskyblue")

def koch(length, depth):

# Base case: a straight line

if depth == 0:

t.forward(length)

return

# Recursive case: four smaller Koch lines, each a third as long,

# with a bump in the middle that points outwards

length = length / 3

koch(length, depth - 1)

t.left(60)

koch(length, depth - 1)

t.right(120)

koch(length, depth - 1)

t.left(60)

koch(length, depth - 1)

# Start at the top left corner of a triangle centred on the screen:

# its middle is a third of the way down from the top edge

side = 340

height = side * math.sqrt(3) / 2

t.penup()

t.goto(-side / 2, height / 3)

t.pendown()

# Three Koch lines, turning right after each one, make the snowflake

t.begin_fill()

for _ in range(3):

koch(side, 4)

t.right(120)

t.end_fill()

t.hideturtle()

turtle.done()

Open in the editor

How the code works

A Koch line

At depth 0 a Koch line is a straight line. Otherwise it is four Koch lines a third as long, and the turns left(60), right(120), left(60) lift the middle into a pointed bump. Every smaller line does the same, so the bumps get bumps.

def koch(length, depth):
    # Base case: a straight line
    if depth == 0:
        t.forward(length)
        return

    # Recursive case: four smaller Koch lines, each a third as long,
    # with a bump in the middle that points outwards
    length = length / 3
    koch(length, depth - 1)
    t.left(60)
    koch(length, depth - 1)
    t.right(120)
    koch(length, depth - 1)
    t.left(60)
    koch(length, depth - 1)

Centre the triangle

The snowflake is built on a triangle with sides of 340. Its middle is a third of its height below the top edge, so starting at (-side / 2, height / 3) puts the finished snowflake in the centre.

# Start at the top left corner of a triangle centred on the screen:
# its middle is a third of the way down from the top edge
side = 340
height = side * math.sqrt(3) / 2
t.penup()
t.goto(-side / 2, height / 3)
t.pendown()

Three sides make a snowflake

Three Koch lines with right(120) between them close the shape. The turtle goes round clockwise, so every bump points outwards, and begin_fill() with end_fill() colours the inside light blue.

# Three Koch lines, turning right after each one, make the snowflake
t.begin_fill()
for _ in range(3):
    koch(side, 4)
    t.right(120)
t.end_fill()

Make it yours

Change the code above and press Run. Some ideas:

Change koch(side, 4) to koch(side, 1), then 2 and 3, to watch each level add smaller bumps.
Delete the t.begin_fill() and t.end_fill() lines to see the outline on its own.
Change t.color("white", "lightskyblue") to t.color("gold", "orange") for a golden star.

What it uses

Every turtle command the site supports is listed on Python turtle online.