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Generating the P3 Tiling

2026-09-08 | original

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Why This Matters

A clear, interactive explainer on how to generate Penrose P3 tilings via recursive subdivision of Robinson triangles. Aperiodic tilings underpin work in quasicrystals, procedural graphics, and non-repeating texture/pattern generation, so accessible algorithmic write-ups lower the barrier for developers and students. The author frames it as groundwork for future posts.

Key Takeaways

Penrose tilings (P1, P2, P3) cover the plane with no translational symmetry; P2 and P3 use just two prototiles.
P3 can be generated by recursively applying four substitution rules to four colored Robinson triangles, each pass yielding finer detail.
The piece is explicitly background material for planned follow-up articles, with sources like Grünbaum and Shephard's Tilings and Patterns.

Worth a Look

Martin Gardner's "Penrose Tiles to Trapdoor Ciphers — If this walkthrough of generating the P3 tiling piqued your curiosity, Martin Gardner's classic collection is the friendliest deep dive into Penrose's kites and darts and why they refuse to repeat. It's written in Gardner's famously clear puzzle-column style, so the aperiodic magic stays fun rather than intimidating.

See Martin Gardner's "Penrose Tiles to Trapdoor Ciphers on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.

Generating the P3 Tiling

2026-06-28

Have you heard of Penrose tilings? Today, I want to describe one algorithm you can use to generate them. My hope is that this article will serve as necessary background material for some future subjects I have in mind, the details of which I'll leave in suspense for now.

What's a tiling, in the first place? Mathematicians call any pattern that covers the 2D plane with no gaps a tiling of the 2D plane. For example, imagine a chessboard extending infinitely in all directions.

The set of shapes that appear in a tiling are called its prototiles. The chessboard tiling uses only one prototile, a square.

A tiling is called periodic if there is a way of shifting the whole plane in such a way that gets you back to the pattern you started at. The chessboard tiling is periodic, because, for example, you can shift the whole thing one square to the right and arrive at an identical pattern. A tiling is called aperiodic if it admits no such translational symmetry.

Penrose tilings were the first known aperiodic tilings of the plane to use only two prototiles. They were discovered by Roger Penrose in the 1970s. There are three variants that I'm aware of, known as P1, P2, and P3. P1 uses four prototiles, while P2 and P3 use only two. We'll concentrate on P3 today.

P1

P2 (Kites and Darts)

P3 (Rhombii)

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Explore topics: penrose tilings p3 tiling aperiodic tiling roger penrose prototiles