Curve S is *n*^{2}, so the product curves on the right side of the spiral are
*n*^{2} – 1,
n^{2} – 4,
n^{2} – 9,
etc.
Curve P is *n*^{2} + *n *so the curves on the left side are
*n*^{2} + *n* – 2,
*n*^{2} + *n* – 6,
n^{2} + *n* – 12,
etc.
These curves have many interesting properties.. I'll point out just a few.
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.
The differences between S and the "S-minus curves" are perfect squares. Similarly, the differences between the pronics and the P-minus curves are pronics. This is another way of saying that these numbers can be factored in the ways we have already described.
Odd curves (those with factors that differ by odd numbers) are offset from curve P, and even curves are offset from curve S.
The *n*th product curve on the even (square) side, counting curve S as zero, has a difference between factors of 2*n* and an offset from curve S of *n*^{2}. The *n*th product curve on the odd side has difference 2*n** *+ 1 and offset *n*^{2 }+ *n*.
I said a moment ago that the blue lines do not represent continuous functions, but obviously I'm using a continuous function to draw them, and of course that function is related to the integer sequences.. I'll explain the formula in a later section but for now, I'll just point out that the number of revolutions made by a blue curve before it straightens out is equal to half the difference between factors. |