GoKawiilA step-by-step algorithm for generating Penrose's P3 rhombus tiling
The piece walks through how to construct the P3 variant of Penrose tilings, an aperiodic pattern that covers the plane using just two rhombus-shaped prototiles. It explains the concept of tilings, periodicity, and how P3 relates to P1 and P2, then presents four substitution rules based on Robinson triangles that can be recursively applied to build increasingly detailed versions of the pattern.
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Penrose tilings remain a foundational example in mathematics for structures that exhibit long-range order without ever repeating, influencing fields from crystallography to computer graphics. Breaking down the generation algorithm into simple recursive rules makes an abstract mathematical curiosity accessible and reusable, which the author signals will support future related work.
- P3 is one of three known Penrose tiling variants and uses only two prototile shapes, both Robinson triangles.
- Aperiodic tilings like P3 cover the plane infinitely without ever repeating under translation, unlike a chessboard grid.
- The tiling can be built recursively by applying four substitution rules that subdivide triangles into smaller ones.
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.Source: k-monk.org, 2026-09-08
Published there as: “Generating the P3 Tiling”
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