Pinwheel Tiling

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

• Store a triangle as three corner points and split it using vertex arithmetic
• Use recursion-style repeated subdivision to turn 4 triangles into 500
• Color shapes by their orientation using atan2

The pinwheel tiling, discovered by Charles Radin and John Conway, is built from a single humble shape: a right triangle with legs 1 and 2 (and hypotenuse √5). Every triangle splits into 5 smaller copies of itself, those split into 5 more, and so on forever. The amazing part: as you keep splitting, the triangles end up pointing in infinitely many different directions — the pattern never settles into a repeating wallpaper.

Welcome to the Pinwheel!

One triangle shape. One splitting rule. Apply it again and again and you get a tiling that mathematicians proved NEVER repeats. Let's build three rounds of it right now!

How It Works

Each triangle is stored as three corner points [a, b, c] with the right angle at b. The subdivide function finds the point h that sits 80% of the way along the hypotenuse from a to c, plus three midpoints — and uses them to cut the triangle into exactly 5 smaller 1:2 right triangles. We run subdivision 3 times on 4 starting triangles, then fill each little triangle with a color chosen from the direction it points.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

def mid(p, q):
    return [(p[0] + q[0]) / 2, (p[1] + q[1]) / 2]

def subdivide(tri):
    a, b, c = tri
    h = [a[0] + 0.8 * (c[0] - a[0]), a[1] + 0.8 * (c[1] - a[1])]
    m1 = mid(a, h)
    m2 = mid(h, b)
    m3 = mid(a, b)
    return [[b, h, c], [a, m1, m3], [m1, h, m2], [m3, m2, b], [m2, m3, m1]]

levels = 3
tris = [[[-190, -190], [190, -190], [190, 0]],
        [[190, 0], [-190, 0], [-190, -190]],
        [[190, 0], [-190, 0], [-190, 190]],
        [[-190, 190], [190, 190], [190, 0]]]
for level in range(levels):
    new = []
    for t in tris:
        new.extend(subdivide(t))
    tris = new

What Do You Think Will Happen?

We start with 4 big triangles, and every round of subdivision replaces each triangle with 5 smaller ones. After 3 rounds, how many little triangles will be on screen? (Hint: 4 × 5 × 5 × 5.) Guess, then run it!

Try It Now

▶ Run ↻ Reset

500 triangles — 4 × 5³. Were you right? Notice how the top half (built from mirrored starting triangles) even picks up different colors, because its triangles point in different directions.

How It Works

The key to the subdivision is the point h, the foot of the altitude dropped from the right-angle corner b onto the hypotenuse. For a 1:2 right triangle, h always sits exactly 80% of the way from a to c — that single fact splits off one perfect small copy ([b, h, c]). The leftover piece is a double-size copy, which the three midpoints slice into 4 more. Because the small copies are tilted by the odd angle atan(1/2) ≈ 26.57° (not a "nice" divisor of 360°), repeated subdivision keeps creating brand-new directions — that's what makes the pinwheel tiling aperiodic.

Explanation Table

Line	What it does
h = [a[0] + 0.8 * (c[0] - a[0]), ...]	Finds the split point 80% along the hypotenuse
return [[b, h, c], ...]	Replaces one triangle with its 5 smaller copies
new.extend(subdivide(t))	Collects all the pieces from this round of splitting
ang = math.degrees(math.atan2(...))	Measures which direction a triangle points, for coloring

Learning Check

Your Turn — Go One Level Deeper

This version starts from just ONE big triangle and subdivides only levels = 2 times, so you can see the 5-way split clearly. Change it to levels = 3. How many triangles will 1 × 5 × 5 × 5 give you?

▶ Run ↻ Reset

levels = 3 gives 125 triangles (5 × 5 × 5). Each extra level multiplies the count by 5 and shrinks every side by a factor of √5.

Experiments

1. Zoom back out. In the first lab, change levels = 3 to levels = 2. You'll know it worked when only 100 chunky triangles fill the square and you can trace each 5-way split by eye.

2. New paint set. Replace 'gold' with 'crimson' and 'royalblue' with 'midnightblue'. You'll know it worked when those colors swap everywhere in the mosaic at once.

3. Finer color buckets. Change (idx // 60) % 6 to (idx // 30) % 6. You'll know it worked when neighboring triangles that used to share a color split into different ones — you're now sorting directions into 30° bins.

4. Half the sky. Delete the last two starting triangles from the tris list. You'll know it worked when only the bottom rectangle is tiled and the top half of the canvas stays white.

Five Copies, Forever!

You just built a genuinely aperiodic tiling — 500 triangles from one splitting rule and a little vertex arithmetic. Mathematicians needed the 1990s to find this one; you drew it in a browser! Up next: Escher-Style Fish Tile — shape a fish so perfectly that copies of it lock together with no gaps.