Tiles: — | φ = 1.6180339...
Penrose tiling via inflation: start with one tile and repeatedly subdivide according to the inflation rules.
Each kite subdivides into 2 kites + 1 dart; each dart into 1 kite + 2 darts — scaled by φ = (1+√5)/2 ≈ 1.618.
The inflation matrix has eigenvalue φ², giving the ratio of kites to darts → φ as depth → ∞.
The tiling is aperiodic: no translation maps it to itself, yet it has perfect 5-fold rotational symmetry and is self-similar. Diffraction pattern shows sharp Bragg peaks with 10-fold symmetry — the signature of a quasicrystal.
Each kite subdivides into 2 kites + 1 dart; each dart into 1 kite + 2 darts — scaled by φ = (1+√5)/2 ≈ 1.618.
The inflation matrix has eigenvalue φ², giving the ratio of kites to darts → φ as depth → ∞.
The tiling is aperiodic: no translation maps it to itself, yet it has perfect 5-fold rotational symmetry and is self-similar. Diffraction pattern shows sharp Bragg peaks with 10-fold symmetry — the signature of a quasicrystal.