Anti-Koch Snowflake

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

• Modify the Koch rule to bump inward instead of outward on alternate levels
• See how one sign change in a fractal rule produces dramatically different shapes
• Understand the concept of rule alternation in recursive systems

While the Koch snowflake always bumps outward, the Anti-Koch alternates: even depths bump out, odd depths bump in. The result is a shape with both spiky tips and concave notches.

Welcome to the Anti-Koch Snowflake!

You've seen the Koch curve bump outward at every level. What if we flipped the direction on every other level? One sign change gives a completely different fractal! Let's code it together!

How It Works

antikoch(n, depth, direction): - direction = 1 → bump outward (left 60°, left 60°, left 60°, then right 120° between) - direction = -1 → bump inward (the angles are negated)

On each recursive call, direction flips: direction * -1.

Sample Code

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

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

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

for _ in range(3):
    antikoch(300, 4, 1)
    monty.right(120)

What Do You Think Will Happen?

The standard Koch always bumps outward, giving a purely spiky shape. The Anti-Koch alternates directions. Will it look more rounded, or more jagged? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

The alternating bumps create a shape with both inward notches AND outward spikes — more complex than the pure Koch because both directions appear. Were you right?

How It Works

60 * direction makes the turn angle positive (left) when direction = 1 and negative (right = inward) when direction = -1. Passing -direction to each recursive call flips the direction at every level, so even-depth segments bump one way and odd-depth segments bump the other.

Explanation Table

Line	What it does
direction parameter	1 = bump outward, -1 = bump inward
monty.left(60 * direction)	Positive direction = left; negative = right
antikoch(n3, depth-1, -direction)	Flip direction at each recursive level
Same structure as Koch	The bump shape is identical — just mirrored

Learning Check

Your Turn — Always Bump Inward

What if you never flipped — always passed -1 as direction? Change antikoch(300, 4, 1) to antikoch(300, 4, -1) and the initial direction to always inward. Predict how it differs from the standard Koch — then run it!

▶ Run ↻ Reset

Always-inward bumps (direction = -1 throughout) creates a Koch snowflake that grows inward — the bumps notch into the triangle instead of radiating outward.

Experiments

1. Compare side by side. Run the standard Koch first (left(60), right(120), left(60)) then the Anti-Koch. You'll know it worked when you can see both shapes.

2. Depth 2 comparison. Try both with depth=2 to see the bumps clearly. You'll know it worked when individual bumps and notches are visible.

3. Change the color. Try pencolor('darkcyan'). You'll know it worked when the curve turns teal.

4. Fill the snowflake. Add begin_fill() before the loop and end_fill() after. You'll know it worked when the interior is filled.

Category 4 Complete!

You've completed 15 fractal patterns — Koch, Sierpiński, Dragon, Hilbert, Barnsley, and more! Fractals show that infinite complexity emerges from simple recursive rules. Up next: Category 5 — Symmetry and Tiling!