Start from a ten-triangle seed and watch an inflation rule grow a patch that can never repeat: thick and thin rhombi assembled from Robinson triangles, exactly the construction Roger Penrose used to break the tiling world's assumption that every repeating-free pattern must have one lurking. Drag to pan, wheel to zoom, scrub the generations and count the ratio of the two rhombi marching — stubbornly — toward φ.
Ten golden triangles meet at the centre — the “star”. It can flip locally to the sun by swapping one ring of five thin rhombi for five thick; such flips are the only way to move between patches.
Each acute golden triangle (36° apex) inflates into one triangle + one gnomon; each gnomon (108° apex) into one triangle + two gnomons — every cut lands on the golden point of its edge, so φ controls the recursion as well as the proportions. The seed triangles alternate orientation; that is the matching rule in disguise. Same-coloured pairs merge into the thick and thin rhombi.
🌀 Never repeating: any tiling by these two rhombi (with matching edges respected) covers the plane but has zero translational symmetry — the “aperiodic set” Penrose published in 1974, after 18 years of attempts by many hands.
🧊 Real matter does it: Dan Shechtman photographed ten-fold diffraction symmetry in a chilled aluminium–manganese alloy in 1982 — quasicrystals, the physical shadow of this construction. He took the 2011 Nobel Prize in Chemistry for being shouted off the conference stage about it.
🌻 Why φ keeps surfacing: inflation multiplies tile counts by the dominant eigenvalue of ⎡2 1⎤⎣1 1⎦ — which is φ² — and the eigenvector pins thick:thin = φ exactly. The ratio in the stats above is the tiling counting its own skeleton.