Aperiodic kites & darts with inflation, deflation, and matching rules
Penrose tiles cover the whole plane without gaps, yet the pattern never repeats. They carry a 5-fold symmetry that no repeating tiling can have.
Press Inflate to split each kite and dart into smaller kites and darts, and Reset to start over. Watch finer detail appear while the 5-fold symmetry stays.
Start with a “sun” of 10 golden triangles and inflate to reveal the aperiodic Penrose P2 tiling. Cyan tiles are half-kites, teal tiles are half-darts.
In the 1970s, Roger Penrose discovered sets of tiles that can tile the plane but only non-periodically -- the pattern never exactly repeats, no matter how far you extend it. These Penrose tilings exhibit 5-fold rotational symmetry, which is impossible in any periodic tiling. They are the mathematical precursor to the discovery of physical quasicrystals, which won Dan Shechtman the 2011 Nobel Prize in Chemistry.
The golden ratio φ = (1 + √5) / 2 pervades every aspect of Penrose tilings: the ratio of tile areas, the frequency of each tile type, and the inflation factor are all powers of φ.
The Penrose P2 tiling uses two shapes: a kite and a dart, both derived from the golden triangle. Press the inflate button to subdivide each tile into smaller kites and darts according to Penrose's substitution rules. Each inflation reveals finer structure while preserving the overall 5-fold symmetry.
Key insight: The ratio of kites to darts in any sufficiently large patch converges to the golden ratio φ ≈ 1.618. This is because the leading eigenvector of the substitution matrix has component ratio φ (its leading eigenvalue is φ² = φ + 1), making the golden ratio an unavoidable feature of the tiling's combinatorial structure.
Use Inflate and Deflate to change the scale, and toggle the matching arcs. The arcs are the edge rules that stop these two rhombi from tiling periodically.
The P3 Penrose tiling uses thick (72°) and thin (36°) rhombi. Toggle the matching-rule arcs to see the constraints that enforce aperiodicity.
The Penrose P3 tiling uses two rhombi: a thick rhombus (72° acute angle) and a thin rhombus (36° acute angle). Matching rules, shown as colored arcs on the tile edges, force the non-periodic arrangement. Without matching rules, the same shapes can tile periodically -- the rules are what make the magic happen.
Key insight: The matching rules are essential. Without them, the thick and thin rhombi can tile the plane periodically. The rules force local configurations that are globally incompatible with periodicity -- a subtle and beautiful interplay between local constraints and global structure.
Adjust the k-range and press Recompute to see the tiling's diffraction pattern. Look for sharp peaks with 10-fold symmetry, which no periodic crystal can produce.
The Fourier transform of Penrose tiling vertex positions reveals sharp Bragg peaks with 10-fold rotational symmetry — the hallmark of quasicrystals discovered by Dan Shechtman in 1982.
The Fourier transform of a Penrose tiling's vertex set reveals sharp diffraction peaks with 10-fold rotational symmetry -- exactly what Shechtman observed in his groundbreaking 1982 experiment with rapidly cooled aluminum-manganese alloys. This "forbidden" symmetry in a diffraction pattern was the first evidence that quasicrystals exist in nature.
Key insight: Sharp Bragg peaks in a diffraction pattern imply long-range order, but not necessarily periodicity. Penrose tilings have "quasiperiodic" order: they can be generated by projecting a higher-dimensional periodic lattice (a 5D cubic lattice) onto 2D. The 10-fold symmetry of the diffraction pattern reflects the symmetry of this higher-dimensional lattice.
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