The Spiral of Theodorus shows a way of constructing every square root of a positive integer:[br][list=1][*]You start with an isosceles right triangle with leg 1.[/*][*]Build another right triangle: its long leg is the previous hypotenuse, and its short leg has length 1.[/*][*]Go to 2.[/*][/list][br]Let's improve this by stepping into 3 dimensions.[br][br]Fiddle with the applet below and catch up with me afterward.

When the black slider, n, is set to 2 or 3, it's easier to see how the green slider affects the diagram, but it's more impressive for large n.[br][br]Conjecture:[br]If all consecutive triangles meet at the same positive angle, the spiral converges toward a particular ray (distance from the middle ray may increase with n, but the angle between the middle ray and the nth hypotenuse approaches 0).[br]If the angles between consecutive triangles are allowed to vary, the spiral could be made to diverge or to converge in any direction we might choose.[br][br]This seems true, but I haven't proved (or implemented) it yet.