I don't understand what you mean by that "pattern occurs in the coordinates of the unit circle". Every number from -1 to 1 occurs as a coordinate of a point on the unit circle. What "pattern" are you talking about?
I was wondering if anything can be done similar to the spiral of theodorus
but obviously since we don't have the pattern occuring in the hypotenuse, I then wondered if it had something to do with 15 degree angle increases but realized that's not such a nice pattern since increasing from 0 to 30 requires two 15 degree increases which produces the same effect on as only 1 15 degree increase from 30 to 45.