Prime Number Spiral

August 25, 2007

This is a neat complex visualization for how numbers relate.
clipped from

Number spirals are very simple. To make one, we just write the non-negative integers on a ribbon and roll it up with zero at the center.

The trick is to arrange the spiral so all the perfect squares (1, 4, 9, 16, etc.) line up in a row on the right side:

Number wheel, figure 1

Numbers on the marked curve are of the form
the famous prime-generating formula discovered by Euler in 1772.

If we continue winding for a while and zoom out a bit, the result looks like this:

Number wheel, figure 1

If we zoom out even further and remove everything except the dots that indicate the locations of integers, we get the next illustration. It shows 2026 dots:

Number wheel, figure 2

Let’s try making the primes darker than the non-primes:

Number wheel, figure 2

The primes seem to cluster along certain curves. Let’s zoom out even further to get a better look. The following number spiral shows all the primes that occur within the first 46,656 non-negative integers. (For clarity, non-primes have been left out.)

Number wheel, figure 2
It looks as though primes tend to concentrate in certain curves
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