Penrose Aperiodic Tiling

P3 rhombus tiling via deflation — never periodic, yet 5-fold symmetric

About

Penrose tilings (1974) tile the plane aperiodically — no translation maps the tiling to itself. Yet they have perfect 5-fold local symmetry and a long-range order detectable by diffraction.
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Computing tiling...

Deflation Rules (P3)

Thick rhombus → 1 thick + 2 thin
Thin rhombus → 1 thick + 1 thin
Scale factor: φ = (1+√5)/2 ≈ 1.618

Both tile types have angles that are multiples of 36° (π/5). The ratio of thick to thin tiles → φ².

Quasi-crystals

Penrose tilings explain quasi-crystals (Shechtman 1984, Nobel 2011) — materials with aperiodic but ordered atomic structure, showing forbidden 10-fold diffraction symmetry.