For those unfamiliar with the Sacks Number Spiral, answering this question will probably take some research, and will probably not be worth the trouble.

For those familiar with the spiral, on this page of Robert Sacks' website, NumberSpiral.com - Product curves
In the details box near the bottom of the page, this bit is stated:

As the S-minus curves extend outward, the angles subtended by the numbers on them approach zero rotations (zero degrees). Therefore the fractional parts of the square roots of integers on those curves must approach zero.


Similarly, the angles subtended by the numbers on Curve P and all the P-minus curves approach an angle of 1/2 rotation (180 degrees). Therefore the fractional parts of the square roots of integers on those curves must approach 1/2.
I can prove the first part by forming a ratio between the squares, lying along the "east" axis, and those offset from the squares, and finding the limit as n approaches infinity. The ratio approaches 1.

The second part is giving me trouble, though. I can't seem to form the right ratio to prove that the fractional parts of square roots on P-minus curves approaches 1/2.

Again, if you're unfamiliar with the sacks number spiral, my question will probably be too much of a hassle, so I'm not exactly expecting too much. Thanks to anyone who gives it a shot, though!