Spiral of Theodorus

Constructing a spiral of square roots

The story of the Pythagoreans discovering, and then attempting to hide, the fact of the irrationality of the square root of 2 is frequently told. Theodorus of Cyrene, a Greek mathematician from the 5th century BC (and possibly a tutor of Plato), is believed to have proven the irrationality of the square roots of the non-square integers up to 17.

He further showed how to construct lines of these lengths via the Spiral of Theodorus. Starting with a 1-1-√2 right angled triangle, the spiral consists of a sequence of right angled triangles, each with one side the hypotenuse of the previous, and a perpendicular second side of length 1.

Pythagoras' theorem tells us that the sequence of hypotenuse lengths is the sequence of square roots of the positive integers. As this activity shows, 17 such triangles are possible before overlapping.