These tilings fill the plane perfectly, forever — yet never repeat. Each tile follows a simple local substitution rule. That local rule is enough to globally forbid periodicity: the pattern has perfect 5-fold symmetry at every scale, yet no translational symmetry at any scale. The gap between local and global shows up here as clearly as it does in emergence, in complex systems, in the distance between a neuron firing and a mind thinking.
P3 rhombus tiling · φ = (1+√5)/2 ≈ 1.618 · Robinson triangle deflation
Roger Penrose discovered these tilings in 1974. The two shapes — a thick rhombus (angles 72° and 108°) and a thin rhombus (angles 36° and 144°) — tile the plane aperiodically when given matching rules. The ratio of thick tiles to thin tiles in any infinite tiling converges to exactly φ, the golden ratio. The tiles encode the golden ratio structurally without any tile knowing about it.
The deflation algorithm subdivides each Robinson triangle into smaller copies according to fixed rules. After enough iterations you have a fragment of the infinite tiling: self-similar in a quasi-crystalline sense — not periodic, not random, but something richer than either.