I've drawn numbers on the angle (shown in green) so we can measure the offsets between the angle and its curves. How do I know these numbers? Because back on page one, when I decided how the spiral is constructed, I said that every number is placed on the angle which equals its square root. (I said it in the form of an equation down in the details box.)
This means that to find the number on any given angle, all we have to do is square the angle.
Of course this is true only if we measure angles in rotations. And that is why I measure angles in rotations on this website, in case you've been wondering.
Let's take the first green dot (moving from right to left) as an example. At that location, the spiral (the gray coil) has made exactly one half of a rotation, so the angle is 1/2. One half squared is one quarter, so the number on that dot is one quarter.
At the next green dot the spiral has made one and a half rotations, so the total angle (not merely its fractional part) at that location is 3/2. If we square that, we get 9/4. That's the number on the second dot. Using the same reasoning for the other dots, the whole sequence goes like this as we travel outward on angle 1/2:
(1/2)2 = 1/4
(3/2)2 = 9/4
(5/2)2 = 25/4
(7/2)2 = 49/4
(9/2)2 = 81/4
(11/2)2 = 121/4
This is a fairly regular pattern, but if you you hold nature to the same high standards as I do, you may find it a bit worrying that all the numerators are odd. Where are the even ones? Have no fear; they will turn up before the end of this page.
Now we can calculate the real offsets. For the first non-zero pronic, 1x2 = 2, the offset is
9/4 – 2 = 9/4 – 8/4 = 1/4
Now that we know that the real offset is 1/4, we can express 2 as the difference between two squares:
2 = 9/4 – 1/4 =
(3/2)2 – (1/2)2 =
(3/2 + 1/2) x (3/2 - 1/2) =
2 x 1
Except for the fact that we're working with fractions, this is exactly what we did with 117 on curve S – 4.
We're almost finished with this topic, but not quite. We still need to round up the missing positive numerators of the pronics.