NumberSpiral

1. Introduction

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:

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

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:

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.)

Details Each non-negative real number n has polar coordinates where theta is the angle measured in rotations, not radians or degrees. One rotation equals 360 degrees. Unless otherwise noted, all angles on this website are measured in rotations. The reason for this will become clear as we go on.

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