Earlier this week I was proofreading a book that mentioned Pythagorean spirals. I had seen them before but never thought about them too deeply, so I gave them a try in GeoGebra. I filled them in with curves of pursuit, just for fun!
To create a Pythagorean spiral, start with a right-angled triangle. Draw another, similar, triangle, rotated around one end of the hypotenuse and scaled so that the side touching the hypotenuse is the same length as the hypotenuse. Then repeat this process with the new triangle. Alternatively, as I’ve done here, rotate in the other direction and scale so that the hypotenuse of the new tringle matches the side of the first triangle that it touches. One option will spiral outwards, the other will spiral inwards.