Aperiodic tiling with 5-fold symmetry via kite-dart substitution rules
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Penrose tiling (1974): aperiodic tiling of the plane using two shapes (kite & dart, or thick & thin rhombuses). Inflation rule: each tile is subdivided into smaller copies of the same two tiles, scaled by φ = (1+√5)/2 ≈ 1.618.
Ratio of kites:darts → φ as level → ∞. The tiling has 5-fold rotational symmetry locally but is never periodic (no translational symmetry).