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)

7. Emergent Patterns

Chaotic mixing is sometimes illustrated by showing repeated folding, or some iterative operation performed on a simple pattern, whereby it becomes increasingly complex. Consider each row of divisors as a periodic signal in time along x. In the divisor plot, the period of each row is equal to y. What happens if the periods are different? Would the same patterns emerge? The principle of sensitivity to initial conditions applies: any slight difference in the period of some combination of these signals results in large differences over time (at large x).

To explore this, a variation of the divisor plot was made which ignores the math and just considers pattern-formation from adjusting the periods. The image at left shows 8 stages of a transformation. The top stage shows a rectangular array of dots. The rows of dots expand rightward, and at different rates – the first row not expanding at all and the last row expanding the most. The difference in expansion among consecutive rows is linear. The final stage, shown at bottom, is the divisor plot.

Three distinct features appear: (1) linear features at the bottom-left (corresponding to the first few zero modulo rays), (2) small linear features at the top that come and go throughout the transformation (corresponding to divisor drips and modulo rays), and (3) various series of parabolas near the middle. The series corresponding to the square root boundary is the most distinct and robust throughout the transformation. Variations using non-linear shifts also produce parabola series. Why are parabolas so common and robust? Might they be attractors?

Click here to see the animation

Want to See More Structure? Visit Two Billion
The image below shows divisors in the range 100,000 to 120,000 at x = 2 billion. Complex patterns can be seen. As we travel to higher numbers, these patterns exhibit more kinds of periodic structure. Perhaps the complexity is related to the hierarchical nature of large composite numbers – which are the products of smaller numbers, some of which themselves are the products of smaller numbers, and so-on – terminating in prime factorization.

Large composite numbers are hierarchical, recursive, and complex.

click here to see
two high-resolution
images...