A non-periodic tiling generated by recursive deflation of two rhombus shapes. The pattern never repeats, yet every finite region appears infinitely often. Roger Penrose described this tiling in 1974 — eight years before Dan Shechtman found the same fivefold symmetry in a real crystal.
5
drag to pan · scroll to zoom · depth 5 ≈ 320 tiles
Two shapes, one grammar. The tiling uses two rhombuses: a thick one (72° corners) and a thin one (36° corners). Every thick tile can be cut into two smaller triangles; every thin tile similarly. Apply this subdivision rule repeatedly and the pattern emerges from nothing but the cuts. At depth 7, there are 1,280 half-tiles filling the canvas.
Fivefold symmetry, no period. The pattern has the same local symmetry as a pentagon — fivefold rotational order — but no translational period. You cannot slide the pattern any distance and have it coincide with itself. Periodic crystals have 2, 3, 4, or 6-fold symmetry; fivefold is forbidden by the mathematics of periodic tilings. When Shechtman saw a fivefold diffraction pattern in 1982, the only conclusion was that the crystal was not periodic. The definition of crystal had to change.