In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
Construction
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 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 nth triangle in the sequence is a right triangle with side lengths √n and 1, and with hypotenuse √n + 1. For example, the 16th triangle has sides measuring 4 (=√16), 1 and hypotenuse of √17
More info on https://en.wikipedia.org/wiki/Spiral_of_Theodorus
Growth rate
Angle If φn is the angle of the nth triangle (or spiral segment), then:
Therefore, the growth of the angle φn of the next triangle n is:
The sum of the angles of the first k triangles is called the total angle φ(k) for the kth triangle. It grows proportionally to the square root of k, with a bounded correction term c2:
where
Radius The growth of the radius of the spiral at a certain triangle n is
A triangle or section of spiral
email me to : muheydari@gmail.com