Divisor Plot

This web page is an expanded version of a presentation given in Boston at the 7th International Conference on Complex Systems

Prime Numbers are the Holes Behind Complex Composite Patterns
Visualizing Number Theory with the Divisor Plot

JJ Ventrella
Jeffrey@Ventrella.com
www.Ventrella.com

5. Relation to the Number Spiral

Below at left is an illustration showing the construction of the number spiral [Sacks 2003]. It was lifted from Sacks' web site. The number spiral is described as follows: roll the number line like a ribbon counterclockwise such that the perfect squares (0, 1, 4, 9, 16...) line up at right. In the middle of the image is my re-creation of Sacks' prime spiral (which is similar to the Ulam Spiral [Stein, 1964]).

It shows a spiral of 4000 numbers with only primes plotted. I decided to try the inverse: composites plotted with gray values determined by the divisor function d(x) for all numbers x in the spiral. This is shown at the right side of the image. White indicates d(x) of 2 (primes), and black indicates d(x) of 20 or more.

Product Curves
Sacks describes a series of what he calls “product curves”, defined by continuous functions that can be mapped onto the spiral. The first curve, shown in the spiral below as a red line, contains the perfect squares. It is called the "S curve". The second curve, shown in orange, traverses the pronics (products of divisors with a difference of 1). It is called the “P curve”. The curve which contains integers that are the products of divisors with a difference of 2 is shown in yellow. It is called the "S-1 curve". The curve which contains the products of divisors with a difference of 3 is called the "P-1" curve. It is shown in black. One more curve, the P-2 curve, is shown in green. Observe the relationship of these curves with the square root parabolas, as shown at right.

The S curve of course corresponds to the square root boundary and traverses the vertices of the square root parabolas, as shown in red. Notice that there are two orange lines corresponding to the P curve, as well as with all other product curves. The S curve is the only one that does not map to a double-line in the divisor plot. Five examples of numbers (24, 25, 28, 29, and 30) have their divisor pairs indicated with colored dots - this is meant to emphasize the fact that Sacks' product curves refer to divisor pairs. Notice that the first 5 numbers on the P-1 curve in the number spiral are prime, and that there is a corresponding gap immediately to the left of the pronics in the divisor plot.

We could continue mapping the product curves from the number spiral onto the divisor plot as shown - forever. It appears that every divisor pair would be visited by a product curve. Let us refer to a product curve on the divisor plot as Cn, where n = 0 corresponds to the square root boundary, and where increasingly higher values of n correspond to pairs of product curves that are increasingly farther away from the square root boundary. An algorithm for specifying Cn (provided by Sacks) is as follows:
y1 = (n+2*i)/2 - n/2 y2 = (n+2*i)/2 + n/2 x = y1 * y2;

....where n is the difference between divisors, and, for each n, i is an integer that varies from 0 to infinity. Another way to describe these curves is with the following equation (also provided by Sacks)...

Notice in the image above that the divisors lying on each product curve increase by 1 as the curve extends to the right. They comprise an ordinal set of integers. We have seen a similar property with two other structures: zero modulo rays, and square root parabolas. This suggests yet another way to organize divisors as sets of ordinal integers in the divisor plot, which we will look into a bit more later.

Rolling the Divisor Plot onto the Number Spiral

The relationships between the number spiral and the divisor plot are numerous. And in fact, Rob Sacks had done preliminary explorations of the divisor plot before working on the number spiral. Since the 1-dimensional number line is coiled-up to occupy a disk (the number spiral), the divisor plot likewise can be rolled up like a scroll to occupy a cylindrical volume, corresponding to the number spiral, as illustrated at right. Notice that the divisor plot is inverted so that the dense area is in contact with the number spiral, as shown.

The figure at right below shows this volume rolled into a number spiral with 20000 numbers. Consider this as a 3D extension of the image above of the divisor function on the 2D number spiral. Vertical density of divisors in the 3rd dimension maps to color density in the 2D image. To help reduce clutter, only divisors less than the square root of x are shown. However, it is still rather dense. And so some variations were tried showing smaller subsets of this set.

This figure shows only conjugate pairs of divisors whose difference is less than or equal to 16. It reveals the fact that the square root boundary, when rolled onto the number spiral, conforms to a cone. It also makes the S-curve and P-curve more visible (they are lying at the left-most and right-most edges of the cone). Notice the similarity to one of Sacks' illustration of product curves:

This figure shows only the divisors for the S-curve less than or equal to the square root of x. Notice that the pattern within the triangle does not occur in the divisor plot. In the divisor plot, these Dx would be spread out and intermittent among many other Dx. The regularities in this pattern show some similarities among square numbers - which I have not explored yet.

This figure shows part of the square root parabola whose vertex is at (10000, 100). It is distorted because it is conforming to a curved surface.

The number spiral is a canvas for visualizing regularities that occur along the number line, which are not easily noticed when viewed along a straight line. Like all things circular and spiral, the number spiral reveals periodic features - which the divisor plot is full of. There are likely more discoveries to be made by plotting divisors on the number spiral in new ways.