It turns out that every integer on a product
curve is intersected by a second related
The existence of the second curve is a
consequence of the following general property: the product of
any two adjacent integers on a product curve can
be found further out on that curve at a
distance from the first factor which is
equal to the first factor's value.
Here's an example:
We can regard the first pair of integers,
4 and 5, as the factors "4 x 5."
To find their product, count "0, 1,
2, 3, 4" starting with the 4. You land
Try again with the second pair, "5 x 8."
Count from 0 to 5 starting with the 5, and
you land on 40.
This works with every pair of integers
on every product curve on the spiral.
As a consequence of this, every integer
on a product curve is intersected by a second
curve that has the factors explicitly present
on it. The intersection of the two curves
is like a visual statement of the multiplication:
the product lies on one curve and the factors
lie on the other. When a curve plays the
second role, I call it a "factor curve."
(Only a small fraction of curves are product
curves, but every curve is a factor curve
with respect to some of the integers on
The relationship between the
two curves at the intersection is interesting.
Let's look at an example: