Koch Snowflake

By the end of this lab you'll be able to:

• Write a recursive function that replaces every line segment with four smaller segments
• Understand what a "base case" is and why it stops the recursion
• See how applying one simple rule repeatedly creates infinite complexity

A six-pointed snowflake with jagged crystalline edges — each edge is itself made of smaller jagged edges, which are made of even smaller ones. Three levels of recursion and a simple rule: bump every segment into a triangular notch.

Welcome to the Koch Snowflake!

The Koch snowflake is one of the most famous fractals. It has infinite perimeter but finite area — impossible to imagine, but easy to code! Let's code it together!

How It Works

The Koch rule for a single line segment of length n: 1. Draw a line of length n/3 2. Turn left 60°, draw n/3 3. Turn right 120°, draw n/3 4. Turn left 60°, draw n/3

At each level, every segment becomes four segments one-third the size. After 4 levels, the edges look smooth and crystalline.

The snowflake is drawn by calling this rule three times, turning 120° between calls to form an equilateral triangle base.

Sample Code

import turtle
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()
monty.pencolor('royalblue')

def koch(n, depth):
    if depth == 0:
        monty.forward(n)
        return
    n3 = n / 3
    koch(n3, depth - 1)
    monty.left(60)
    koch(n3, depth - 1)
    monty.right(120)
    koch(n3, depth - 1)
    monty.left(60)
    koch(n3, depth - 1)

monty.penup()
monty.goto(-150, 90)
monty.pendown()

for _ in range(3):
    koch(300, 4)
    monty.right(120)

What Do You Think Will Happen?

At depth 0, koch(n, 0) just draws a straight line. At depth 1, each segment has a triangular bump. At depth 4, will the edges look smooth or jagged? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

Jagged but beautiful — the edges have a crystalline, snowflake-like appearance. Were you right?

How It Works

Each call to koch(n, depth) with depth > 0 makes four recursive calls to itself with depth - 1. When depth finally reaches 0, the function just draws a line and returns. This is the base case — without it, the recursion would never stop.

The total number of line segments drawn is 4^depth: at depth 4 that's 4^4 = 256 segments per side, or 768 segments for the whole snowflake.

Explanation Table

Line	What it does
if depth == 0: monty.forward(n)	Base case: draw a line and stop recursing
n3 = n / 3	Each recursive segment is one-third the size
monty.left(60)	Turn to form the triangular bump's left side
monty.right(120)	Turn to form the bump's right side (two 60° turns = 120°)
Four koch() calls	Replace one segment with four

What is Recursion?

Recursion means a function that calls itself. The trick is the base case — a condition that stops the calls. Without a base case, the function would call itself forever (and Python would stop it with a RecursionError).

Learning Check

Your Turn — Try Depth 2

Change depth=4 to depth=2 in the koch() calls. Predict how much simpler the snowflake edges will look — then run it to check!

▶ Run ↻ Reset

At depth 2, each edge has only 4^2 = 16 segments — you can clearly see the individual triangular bumps on each bumpy bump.

Experiments

1. Try depth 1. Just one round of bumping — three big triangular notches per side. You'll know it worked when you see a star of David shape.

2. Add fill. Before the loop, add monty.fillcolor('lightblue') and monty.begin_fill(). After the loop, add monty.end_fill(). You'll know it worked when the snowflake is filled with light blue.

3. Change the color. Try pencolor('white') with a dark background: add turtle.bgcolor('navy') before creating monty. You'll know it worked when you see a white snowflake on blue.

4. Draw just one edge. Replace the for loop with a single koch(300, 4) call. You'll know it worked when you see just one Koch curve instead of the full triangle.

Your First Fractal!

You just drew infinite complexity from a 4-line recursive rule! The Koch snowflake has infinite perimeter — every time you zoom in, there are more bumps. Up next: Sierpiński Triangle — the other iconic fractal.