# Sieve of Eratosthenes Project

I like geometric art projects. I especially like playing with projects revolving around prime numbers. One of my favorites is the Sieve of Eratosthenes, which is a simple algorithm for finding prime numbers by crossing out every nth number. Proposed by Eratosthenes of Cyrene approx. 2200 years ago, the idea is that you write out the numbers that you’re interested in, from 1 to whatever (say, 100). First, pick n=2. put your pencil on the number 2, and start counting, crossing out every 2nd number (4, 6, 8, etc.) When you’re done, go back to 3, and cross out every 3rd number (6, 9, 12, 15, etc.) Keep doing this until you get to 11. Each number left uncrossed will be prime (2, 3, 5, 7, 11, etc.). You could speed the process up by only using n=prime numbers (2, 3, 5 and 7).

(Normal Sieve, for primes between 1 and 100, from left to right, top to bottom.)

Normally, people will lay the numbers out in a grid, generally a square, and number the grid points from left to right, top to bottom (as in the example in the wiki article). But, if you’re looking for patterns, this isn’t the best approach, because you can’t resize the grid to add more numbers without renumbering everything. And, the dimensions of your grid (10×50, or 14×35) impose an artificial factoring element that impacts the patterns you get (numbers not divisible by 5, or not divisible by 7). If you’re not sure what I mean, try making two Sieves, one 10×50 and one 14×35, in both number the grid points from 1 to whatever left to right, top to bottom, and compare the patterns for the prime number locations in each.

(Newton article spiral approach.)

Interestingly, the Japanese science magazine Newton had a feature article some months ago on the Riemann Hypothesis, and prime numbers in general, and one of their illustrations was a Sieve, created using the Matlab software, where the numbers were in a continuous spiral from the origin on an x-y graph. There were still no useful patterns, but the display space was now infinitely expandable – just add more graph paper. I decided I’d try something similar, but snaking back and forth only in the positive x-y quadrant.

(My approach.)

My project this time consisted of making a grid on a piece of matte board 36×36 squares, numbering the squares, and then applying the sieve algorithm with little pieces of colored construction paper. It took 3 days, several large sheets of paper and a full tube of wood glue to finish. Each pass of the sieve (n=2, 3, 5, 7, 11, etc.) is a different color paper, up to n= 19 or so, where it was easier to just repeat the colors. One pattern I was hoping would emerge is that certain numbers would get these big stacks of paper because they have so many factors in common with each other. But, even the heights of the stacks failed to produce interesting textures. Oh well.

Anyway, I stitched the photos of the matte board together in Movie Maker and added a sound track using the Rockit 8-bit synth. In part, one of the reasons for posting the project as a youtube video was to show off what you can do with the Rockit.