Spiral of Theodorus

In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral) is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.

The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle is formed, an automedian right triangle with one leg being the k hypotenuse of the prior triangle (with length √2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is √3. The process then repeats; the i th triangle in the sequence is a right triangle with side lengths √i and 1, and with hypotenuse √i + 1. For example, the 16th triangle has sides measuring 4 (=√16), 1 and hypotenuse of √17

Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.

Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.

Each of the triangles' hypotenuses h_i gives the square root of the corresponding natural number, with h₁ = √2.

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