Penrose tilings are non-periodic tilings of the plane discovered by Roger Penrose in the 1970s. The P3 variant uses two rhombus shapes — a thick rhombus (72°/108° angles) and a thin rhombus (36°/144° angles) — to tile the plane aperiodically while maintaining 5-fold rotational symmetry.
1/φ).φ = (1 + √5) / 2 ≈ 1.6180339887 Tile angles: Thick: α = 72°, β = 108° Thin: α = 36°, β = 144° Deflation scale: s = 1/φ per iteration Tile ratio: N(thick)/N(thin) → φ as iterations → ∞ Diffraction: Bragg peaks with 10-fold symmetry, indexed by Z⁴ lattice
In 1984, Dan Shechtman discovered a physical aluminium-manganese alloy whose X-ray diffraction pattern showed 10-fold symmetry — impossible in conventional crystallography. He was initially ridiculed, but the discovery of quasicrystals earned him the 2011 Nobel Prize in Chemistry. The mathematical blueprint for quasicrystals is precisely the Penrose tiling.
A Penrose tiling is a non-periodic tiling of the plane discovered by mathematician Roger Penrose in the 1970s. It uses a small set of tile shapes that can cover an infinite plane without ever repeating the same pattern, yet every finite region appears infinitely often.
The P3 Penrose tiling uses two rhombus shapes: a thick rhombus with angles 72° and 108°, and a thin rhombus with angles 36° and 144°. The ratio of thick to thin rhombuses in any valid P3 tiling approaches the golden ratio φ ≈ 1.618.
Deflation is a substitution rule where each tile is replaced by smaller tiles: a thick rhombus splits into one thick and two thin rhombuses; a thin rhombus splits into one thick and one thin rhombus. All new tiles are scaled by 1/φ compared to the original.
The angles in Penrose tiles are multiples of 36° = π/5, the fundamental angle of a regular pentagon. The construction begins from a decagon (10-fold symmetric arrangement) and all deflation operations preserve the 5-fold rotational structure.
A quasicrystal is a physical material with ordered but non-periodic atomic arrangements, showing diffraction with forbidden symmetries like 5-fold or 10-fold. The Penrose tiling is its 2D mathematical model. Dan Shechtman discovered physical quasicrystals in 1984 and won the 2011 Nobel Prize in Chemistry.
The golden ratio φ = (1+√5)/2 ≈ 1.618 appears throughout: the tile edge ratios involve φ, the ratio of thick to thin tiles approaches φ, the inflation factor between deflation levels is exactly φ, and the tile angles are multiples of 36° whose cosine involves φ.
Yes. A tiling is aperiodic if it has no translational symmetry — you cannot shift the pattern by any non-zero vector and have it match itself. Penrose proved his tiles admit no periodic tilings, even though they cover the entire infinite plane.
There are uncountably many distinct Penrose tilings of the infinite plane. Although any finite region of one tiling appears in every other Penrose tiling (local indistinguishability), the global arrangement is never the same.
The diffraction pattern of a Penrose tiling shows sharp Bragg peaks with 10-fold symmetry — exactly what is observed in physical quasicrystals. The pattern is purely discrete but indexed by a 4-dimensional lattice Z⁴ rather than a 3-dimensional one.
Yes. The Penrose P3 tiling can be obtained by projecting a strip of the 4-dimensional integer lattice Z⁴ onto a 2D plane at a slope related to the golden ratio. This 'cut-and-project' method connects Penrose tilings to higher-dimensional crystallography.
Penrose tilings are aperiodic tilings of the plane discovered by Roger Penrose in 1974 — they cover the plane completely with no gaps or overlaps, but unlike periodic tilings (such as square or hexagonal grids) they never exactly repeat under any translation. The most common variants are P2 (kite and dart) and P3 (thick and thin rhombi), both of which exhibit exact 5-fold rotational symmetry. The ratio of the two tile types converges to the golden ratio φ ≈ 1.618 as the tiling grows, and the structure is connected to the diffraction patterns of quasicrystals — a discovery that earned Dan Shechtman the 2011 Nobel Prize in Chemistry.
This simulation generates P3 rhombus tilings via the deflation substitution rule, letting you zoom into successive inflation levels and verify the 5-fold symmetry. The matching-rule arrows on tile edges demonstrate why no periodic arrangement satisfying the rules is possible.
What makes a Penrose tiling aperiodic?
A tiling is aperiodic if no translation vector exists that maps the tiling to itself — the pattern never exactly repeats. Penrose tilings achieve this through local matching rules on the tile edges (arrows that must align), which globally forbid periodicity while still permitting the plane to be tiled completely. The proof uses the inflation/deflation self-similarity: if the tiling were periodic, it would remain periodic after deflation, but deflation reduces the tile size by φ⁻¹ indefinitely, producing a contradiction.
What is the golden ratio's role in Penrose tilings?
The golden ratio φ = (1 + √5)/2 ≈ 1.618 appears throughout Penrose geometry. In the P3 tiling the angles of the thick rhombus are 72° and 108°, and those of the thin rhombus are 36° and 144° — all multiples of 36° = π/5, directly related to the regular pentagon. The ratio of thick to thin rhombi in a large Penrose tiling converges to φ : 1 ≈ 1.618 : 1. The deflation rule scales all lengths by φ⁻¹ at each step.
What are quasicrystals and how do they relate to Penrose tilings?
Quasicrystals are solid-state materials whose atomic arrangements display 5-fold (or other non-crystallographic) rotational symmetry combined with long-range order but no translational periodicity — the 3D analogue of a Penrose tiling. Dan Shechtman discovered the first quasicrystal (an aluminium-manganese alloy) in 1982 and published the finding in 1984; it was initially so controversial that his supervisor asked him to leave the research group. He received the 2011 Nobel Prize in Chemistry for the discovery.
Each thick rhombus is replaced by two thick and one thin rhombus scaled by φ⁻¹; each thin rhombus is replaced by one thick and one thin rhombus at the same scale. After k deflation steps the number of tiles grows as φ^k (approximately), maintaining the φ : 1 ratio between thick and thin tiles. The substitution is self-similar: a region of a deflated tiling looks like a scaled-up version of the original, confirming the fractal-like structure.
Yes — there are uncountably many distinct Penrose tilings of the plane, all sharing the same local statistical properties (the same frequencies of every finite patch). Any two Penrose tilings are locally indistinguishable: every finite region of one tiling appears infinitely often in every other Penrose tiling. This property, called local indistinguishability, makes the global aperiodicity invisible from any local patch — you can never determine which specific Penrose tiling you are in by examining only a finite region.
Yes — all Penrose tilings can be constructed by projecting a 2D slice of a 5D hypercubic lattice onto the plane. The 5D lattice is periodic, and the 2D projection is at an irrational angle to all lattice directions, which destroys periodicity while preserving the 5-fold symmetry of the 5D lattice. This "cut-and-project" or "slice" method provides a clean algebraic framework and explains why quasicrystals show sharp diffraction peaks (the 5D structure is perfectly periodic) despite lacking translational periodicity in 3D.
The P2 Penrose tiling uses two quadrilateral tiles derived from a regular pentagon: the "kite" (a quadrilateral with angles 72°, 72°, 72°, 144°) and the "dart" (with angles 36°, 72°, 36°, 216°). Both have side lengths in the ratio φ : 1. Matching rules (usually shown as arcs that must align) prevent periodic tilings, and the kite-to-dart ratio converges to φ². P2 and P3 tilings are dual to each other and mutually locally derivable.
A Conway worm (named after John Conway) is a chain of thick and thin rhombi in a Penrose P3 tiling that forms a straight or near-straight strip running across the tiling. Conway showed that every Penrose tiling can be analysed using five families of such strips (called Ammann bars), whose spacings follow the Fibonacci sequence with long (L) and short (S) gaps in the ratio φ : 1. The intersection pattern of these five strip families encodes the global structure of the tiling.
Yes — in 1997 Roger Penrose and Pentaplex Ltd filed a lawsuit against Kimberly-Clark, the manufacturer of Kleenex quilted toilet paper, for embossing their product with a Penrose tiling pattern without a licence. The case was settled out of court. Penrose holds patents on his tiling patterns (the UK patent GB2010900 expired in 1997), and the incident raised interesting questions about whether mathematical patterns can be copyrighted or patented.