Penrose Rhombus Tiling

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

• Use complex numbers as 2D points to do geometry with plain arithmetic
• Apply two recursive subdivision rules (thin → 2 pieces, thick → 3 pieces)
• Explain why the golden ratio makes this tiling never repeat

In the 1970s, Roger Penrose discovered two rhombus shapes — one thick, one thin — that can tile a floor forever without the pattern ever repeating. This program builds his famous P3 tiling by "deflation": start with a sun of 10 golden triangles around the center, then repeatedly split every triangle into smaller golden triangles using the golden ratio φ ≈ 1.618. Four rounds later, gold and blue rhombi bloom into stars and suns across the whole canvas.

Welcome to the Tiling That Never Repeats!

Two rhombus shapes, two splitting rules, one magic number: φ. Slide this pattern in any direction and it will NEVER line up with itself. Let's grow it from ten triangles!

How It Works

Every triangle is stored as [kind, a, b, c] where the corners are complex numbers — complex(x, y) acts like a 2D point, and a + (b - a) / phi finds the point that cuts side ab in the golden ratio. A thin half-rhombus splits into 2 smaller triangles; a thick one splits into 3. After subdividing 4 times, we fill each triangle (thick = gold, thin = royal blue) and then trace only the two outer edges c → a → b, so each pair of triangles displays as one complete rhombus.

Sample Code

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

phi = (1 + math.sqrt(5)) / 2

triangles = []
for i in range(10):
    a1 = (2 * i - 1) * math.pi / 10
    a2 = (2 * i + 1) * math.pi / 10
    b = complex(math.cos(a1), math.sin(a1)) * 195
    c = complex(math.cos(a2), math.sin(a2)) * 195
    if i % 2 == 0:
        b, c = c, b
    triangles.append(['thin', complex(0, 0), b, c])

levels = 4
for level in range(levels):
    new = []
    for kind, a, b, c in triangles:
        if kind == 'thin':
            p = a + (b - a) / phi
            new.append(['thin', c, p, b])
            new.append(['thick', p, c, a])
        else:
            q = b + (a - b) / phi
            r = b + (c - b) / phi
            new.append(['thick', r, c, a])
            new.append(['thick', q, r, b])
            new.append(['thin', r, q, a])
    triangles = new

What Do You Think Will Happen?

Every round, each thin triangle becomes 2 pieces but each thick one becomes 3. After 4 rounds of that, which color do you think will cover more of the picture — gold (thick) or royal blue (thin)? Guess, then run!

Try It Now

▶ Run ↻ Reset

Gold wins. After 4 rounds there are 210 thick half-rhombi and only 130 thin ones — and each thick rhombus is fatter too. (Fun fact: the thick-to-thin ratio approaches exactly φ as you keep subdividing.) Were you right?

How It Works

The two triangle kinds are the halves of Penrose's two rhombi: the thin rhombus (36° corners) and the thick rhombus (72° corners). Golden-ratio cuts like a + (b - a) / phi split each one into perfect smaller copies — that only works because a golden triangle's sides are in ratio φ : φ : 1. Because φ is irrational, the numbers of thick and thin tiles can never settle into a neat whole-number ratio, so no shifted copy of the pattern can ever match itself: the tiling is aperiodic. Notice the 5-fold symmetry everywhere — the "forbidden symmetry" that ordinary repeating wallpapers can't have. When scientists found this same 5-fold pattern frozen in a metal alloy in 1982 (they named them quasicrystals), it won Dan Shechtman the Nobel Prize.

Explanation Table

Line	What it does
phi = (1 + math.sqrt(5)) / 2	The golden ratio, 1.618... — the engine of the whole tiling
b = complex(math.cos(a1), math.sin(a1)) * 195	Places a corner on a circle using a complex number as an (x, y) point
p = a + (b - a) / phi	Cuts the side ab at the golden ratio point
new.append(['thick', p, c, a])	Adds one of the smaller triangles that replaces the old one

Learning Check

Your Turn — Deflate Deeper

This version stops at levels = 2, so you can still see the ten big wedges of the starting sun. Change it to levels = 4. Each round shrinks the rhombi by a factor of φ — will the pattern at the center still look like a sun when you're done?

▶ Run ↻ Reset

Yes — a smaller sun of ten gold rhombi survives at the very center, now ringed by blue stars. Deflation shrinks the pattern but the sun motif keeps reappearing at every scale.

Experiments

1. One more round. In the first lab, change levels = 4 to levels = 5 (give it a few seconds — that's 890 triangles). You'll know it worked when the rhombi shrink again and new five-pointed stars appear that weren't visible before.

2. Night mode. Swap the colors: make thin 'gold' and thick 'midnightblue'. You'll know it worked when the tiling flips to mostly dark with golden star outlines.

3. Show the hidden triangles. In the edge-drawing loop at the bottom, add monty.goto(c.real, c.imag) as a final line inside the loop. You'll know it worked when a seam appears across each rhombus, revealing the two triangles it is really made of.

4. Shrink the sun. Change both * 195 to * 120. You'll know it worked when the whole decagon pulls toward the center with white space around it.

Forbidden Symmetry Unlocked!

Two shapes, two rules, and the golden ratio — and you drew a pattern that never repeats, with the 5-fold symmetry crystals were never supposed to have. Penrose proved it; you programmed it! Browse the Turtle Graphics Gallery for your next challenge!