by Jeffrey Ventrella

In the divisor plot, relatively dense strings of divisors can be seen beneath some numbers, like dripping water. I call these "divisor drips". That's just a poetic name for numbers that have lots of divisors (Dx), such as the highly composite numbers (numbers that have more divisors than any lesser number).

On either side of dense divisor drips you can often detect radiating lines projecting outward and downward. I call these "reflection rays". These are reflections of the divisor drip, equivalent to addition or subtraction of multiples of y to the y coordinates of divisors. Reflection rays are a natural outcome of dense divisor drips. You could also think of the reflection rays associated with a particular divisor drip Dx as the set of common multiples that x shares with other numbers.

The illustration below shows D0, and the first 4 of its positive reflection rays, indicated with arrows. Also shown are a few of the reflection rays (y<5) of D12, indicated with thin lines. Notice that there is another clearly-visible divisor drip, similar to the one at D12, a bit farther to the right. Just take a guess at what that number is without counting up to that location. If you guessed that it is 24, you would be correct! With this small range of the number line, it is not hard to guess the numbers simply by looking at the divisor drips (or the absence of them).

In D24, you may have noticed the divisor 8 in the drip, which you may have used as your clue. You might have also noticed the '5' sticking out just to the right of the D24 drip. It is a lonely divisor: the square root of 25. Finally, you can probably easily guess what the last two numbers in this graph are without actually counting up to them.

A kind of portraiture of numbers is revealed as we study this picture. If you study it long enough, you might start to detect certain social trends, such as the way twin primes often snuggle up to either side of strong composite numbers.

The image below shows x at 12! (or 12 factorial, which is equal to 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12, or 479,001,600). It has a strong divisor drip and clearly-visible reflection rays.

Since every integer could be considered as a divisor of 0 (when multiplied by 0), 0 is considered here as the ultimate highly composite number - having a completely solid divisor drip, and subsequent reflection rays. There are infinitely many zero reflection rays Z numbered n from 1 to infinity. Zero reflection 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, and only one.

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. (It has many other divisors as well). The right half of the image shows that the divisor pattern at 12! is identical to the pattern at 0 for all y less than 13.

In the divisor plot, any x = n! will look like the zero region among the first n divisors. Let's think bigger now, and consider the number 25! For the first 25 divisors, the region in the divisor plot near this number looks identical to the region near zero.

The bigger the factorial number, the more its neighborhood looks like the neighborhood of 0. I suspect that not only the immediate region of the n! but the entire pattern looks identical to 0, because it is a repeating pattern, a pattern that repeats itself every n! numbers. To explore this idea, below are some examples of patterns of 3!, 4!, and 5! In each case, the solid divisor drip of length n repeats itself every n! number. However, there may be other repetitions of n-length divisor drips. For instance, notice the repetition of a length-4 divisor drip at 12, which occurs before 4! In the illustration below. Also, not shown here is a repetition of a 5-length divisor drip at 60, which occurs before 5!

A number like 25! Is not very interesting when viewed as a one-dimensional string of digits (as shown here in base 10):

15,511,210,043,330,985,984,000,000

But there is hidden structure in this number, as revealed in the divisor plot (and possibly other visual representations). Compare this to the cells in a grasshopper. If you could line the millions of grasshopper cells in a row, it would be a long row indeed. Impressive in sheer length, but not very impressive otherwise. Only when the cells are arranged in the form of a grasshopper can we appreciate the beauty and function of this insect. Most importantly, a single string of cells is not capable of jumping over a mushroom.

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 illustration below shows 7! with such a sequence highlighted. The sequence is in fact longer than 7, as indicated here by the divisors 8, 9 and 10 in the first positive reflection ray.

What is now coming into focus is a two-dimensional portrait of a factorial number, or more accurately: the number plus its immediate neighborhood. As viewed in the divisor plot, the divisor drip of a factorial n! has no gaps among the first n divisors. But there is another interesting feature of this factorial portrait. The associated reflection rays (specifically the first set on either side) each represent a contiguous set of composite numbers. In the example of 7! we could just as easily apply the same principle to the numbers to the left-side of the divisor drip: those numbers are also composite.

The two numbers immediately to the left and to the right of n! are special, however. They may or may not be composite numbers.

The illustration 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. 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.

It is easy to compare the first 3 or 4 rows in the divisor plot to musical polyrhythm. 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. 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. However, classical Western, African, and Indian music sometimes incorporates small prime number beats such as 5 or 7, juxtaposed against 4 or 8. 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.

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 reflection ray (Zn). We can also classify them as existing in rows (seen as periodic signals moving from left to right). 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 even more ways to classify divisors, and they are much more interesting, visually and mathematically.

Next chapter: 3. The Square Root Spine