Spiral of Theodorus

The Spiral of Theodorus (also called the square root spiral, or Pythagorean spiral) is built from a chain of right triangles, each one sharing a leg with the next. The first triangle has two legs of length 1, giving a hypotenuse of $\sqrt{2}$; every following triangle uses the previous hypotenuse as one leg and adds a new leg of length 1, so the $n$th hypotenuse has length exactly $\sqrt{n}$. It is named after the Greek mathematician Theodorus of Cyrene, who is said to have used this construction around 400 BC to show that the square roots of the non-square integers from 2 to 17 are irrational. Unlike the Archimedean or logarithmic spirals above, it has no single smooth formula — it is built triangle by triangle, and the familiar spiral shape only emerges once enough triangles have accumulated.