Drag the tiling to pan once you've zoomed in.
About this
Cut a regular pentagon's golden triangles in half and you get two shapes: a thin kite half and a thin dart half, both made of two isosceles triangles whose long side is exactly φ times their short side, where φ = (1+√5)/2 ≈ 1.618. Robinson and Penrose found a single rule for splitting each one into smaller kite-halves and dart-halves, all scaled down by 1/φ: every kite-half becomes one kite-half plus one dart-half; every dart-half becomes one kite-half plus two dart-halves.
Apply that rule to a ring of ten kite-halves and the result is the famous Penrose tiling: a pattern built from only two tile shapes that can cover the plane completely, yet never falls into a repeating, shiftable pattern the way square tiles or hexagons do. Any finite patch of it recurs infinitely often elsewhere in the tiling — just never on a regular grid.
Run the substitution a few more times and a second thing falls out for free: because each kite-half turns into 1 kite-half and 1 dart-half, and each dart-half turns into 1 kite-half and 2 dart-halves, the ratio of darts to kites after each generation is governed by the same recurrence that defines φ itself. It doesn't just resemble the golden ratio — it converges to it exactly, and you can watch the approach above sharpen generation by generation.