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)

2. Divisor Drips and Modulo Rays

Relatively dense strings of divisors Dx can be seen beneath some x, like dripping water, as seen in the image below at right. These are called “divisor drips”. That's just a poetic name for Dx that have lots of divisors, such as the highly composite numbers (numbers that have more divisors than any lesser number).

On either side of these divisor drips you can detect radiating lines projecting outward and downward, called “modulo rays”. These are reflections of the divisor drip, equivalent to addition or subtraction of multiples of y to the y coordinates of divisors.

Modulo rays are a natural outcome of dense divisor drips. Since every integer is a divisor of 0, it is considered here as the ultimate highly composite number – having a completely solid divisor drip, and subsequent modulo rays. The image below at left shows D0 and the first 4 of its positive modulo rays. A few of the modulo rays (y<5) of D12 are indicated with thin lines. The image at right shows x at 12! (12 factorial, equal to 479,001,600). It has a strong divisor drip and distinct modulo rays.

Definition of Zero Modulo Ray
There are infinitely many zero modulo rays Z numbered n from 1 to infinity. Zero modulo ray Zn is defined as the ray originating at (0,0), and having slope 1/-n. Each Zn has an ordinal set of divisors lying on it The higher the value of n, the more sparse the divisors along the ray. Every divisor in the divisor plot is a member of one Zn.

Why is Zero like 12 Factorial?
If we allow the divisor plot to include negative x, we get the pattern shown below at left. We know that 12! has 1 through 12 as divisors, but it has many others as well. The image below at right shows that the divisor pattern at 12! is identical to the pattern at 0 for all y < 13. In the divisor plot, any x = n! will look like the zero region among the first n divisors.

Number Segments with No Primes
One way to find an arbitrarily long contiguous series of composite numbers on the number line is to choose a factorial n! The numbers n!+2, n!+3, n!+4...n!+n comprise a contiguous sequence with no primes. The figure at right shows 7! with such a sequence highlighted. The sequence may in fact be longer, as indicated here by the divisors 8, 9 and 10 in the first positive modulo ray.

Patterns in Time
The image below shows an imaginary scene with 6 objects with regularly-blinking lights. They start lined-up at the left of a track, move to the right at a constant speed, and then stop at the right side. A camera takes a timed-exposed photograph while the objects are moving across the track.

If object 1 blinked once every second, object 2 blinked once every 2 seconds, object 3 once every 3 seconds, and so-on, and if they all started blinking at the same time at the start, then the resulting timed-exposure photograph would be an exact replica of the divisor plot pattern.

Musical Polyrhythms
It is easy to compare the first 3 or 4 rows in the divisor plot to musical polyrhythm (also irrational rhythm). For instance, a jazz, rock, or traditional African rhythm might juxtapose periods of 2 and 3, combined in various ways to create composite periods of 6, 12, etc. Since periods of 2 and 3 (and their multiples) come naturally to the ear (and to dancing feet) we rarely encounter 5, 7, or other prime number periods in popular music.

Rows
As we have just seen, it is useful to classify divisors in ways besides being members of a vertical divisor drip (Dx), or as members of an angled zero modulo ray (Zn). We can also classify them as existing in rows (seen as periodic signals along x). This is one way to visualize the Sieve of Eratosthenes. Let's refer to the horizontal rows that contain divisors as R. It is equivalent to the integer values of y. Later, we will see that there are more ways to classify divisors.

Here is a diagram showing a 2 against 3 polyrhythm:

Notice that each down-beat (when both rhythms have an X in the box) creates a miniature divisor drip, and that on either side are empty spaces - analogous to twin primes on the number line.