Spiral of Theodorus 3D
The Spiral of Theodorus shows a way of constructing every square root of a positive integer:
- 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.
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.
Conjecture:
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).
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.
This seems true, but I haven't proved (or implemented) it yet.