A First Hypothesis on My “Nixon” Prime Spirals

A 'Nixon' SpiralSomewhere around 200BC a Greek mathematician called Eratosthenes described a sieve method that could be used to generate prime numbers. For low prime numbers it’s not too hard to tell whether a number is prime by testing it for having integer factors. But as you move on to large and larger numbers this takes more and more time, so methods based on this sieve are often used to gain speed. It works like this:

Write down all the numbers you want to test for primality. Once you have this list locate the first prime number after 1 and erase every multiple of it. This will remove 4, 6, 8, and all other even numbers, leaving 2. Having done this locate the next unerased number, which will be 5. So now count along your list and erase every multiple of it such as 10, 15, 20, and so on. Now move onto the next number, which will be 7, and delete all multiples of that (14, 21, 28, and so on), and the next unerased number will be 11.

Sieve of EratostheneseYou see what’s happening here? One by one we are discovering the prime numbers as we move up the number list. And at the same time, we’re are quickly removing rafts of non-prime numbers throughout the rest of the list, consequently discovering primes without a single division operation in sight.
Now, imagine that you write this list of numbers in a spiral and then draw lines instead of erasing numbers when you conduct each sieve operation. Depending on how closely together you wrote the numbers, you’ll see spirals radiating out in a variety of different curves.

Considering the prime spirals I’ve been creating recently, in which only prime numbers (or pairs, triplets or other combinations of primes) are plotted, you can imagine the primes not plotted as being drawn using the sieve process in white lines.  And this is what I hypothesize I am finding. These spirals actually bring out the sieve spirals, resulting in the patterns found.

A Prime Spiral with radial 'spokes'Of course, when the correct spacing values are used the spirals form straight radial lines, and that’s why the images in my previous article look more like bicycle wheels than spirals – I used multiples of 360 in the distance to straighten them up. So, to remove this straightening effect I removed the 360 and replaced it with prime numbers and found spirals going in both clockwise and counterclockwise directions (at the same time), and with different curvatures too.

To get a better understanding of what is happening I then wrote the code to animate all distances between potential plot points from pπ to π ÷ 2, π ÷ 3, π ÷ 5, π ÷ 7,  and so on (diving pi only by prime numbers to remove the possibility of integer divisors creating artificial patterns). What happens when π is used is you get a straight line because the distance between all potential plots is a semi-circle, so the spiral is a single line going straight out. as you increase the divisor, more ‘spokes’ appear as more potential plot locations become available, until you get into divisors of a few hundred, when the spokes are so close to each other patterns can start to be revealed.

Interestingly, if this process continues, the final result will be a single straight line again. But only when it reaches π/infinity. So in the programs below, once the divisor gets into the several thousands, you’ll see a new spiral begin to emerge and the prime patterns will be harder to see, This is because the distance between potential plots is so close that plot points are overlapping each other, resulting in just a spiral line (the one that will end up straight at infinity).

So following is what happens when you use (for example), the prime number 2143 as the divisor for calculating the plotting distance gap:

A 'Nixon' Spiral

Here we see another benefit to not making the ‘spokes’ straight, because now it’s possible to determine the directions in which they curve. Anyway, for comparison (and to try to check that all this isn’t hogwash and caused by simple interference patterns or something), here’s the same value used but with random numbers ending in 1, 3, 7, or 9 being plotted, rather than prime numbers – not much to see here (thankfully!):

A Random Spiral

Here is a combined program you can try for yourself to create prime and random ‘Nixon’ Spiral animations. Note that things are a little boring for the first few dozen frames, but bear with the animations, as patterns very soon start to emerge – there are also a number of keyboard options available for controlling the animation:

By the way, after viewing these animations over and over, I’m now fairly sure that above a rough value of around 1700 or thereabouts, the interference effects (unexpected white ‘spokes’) of non-prime plot distance gaps is vastly reduced. So here’s a version of the Prime Spiral Animation program that counts in linearly decreasing gap sizes, starting with π ÷ 1700 (then π ÷ 1701, π ÷ 1702, and so on), rather than by the next smallest prime number fraction of π ÷ n. When you watch this animation you can see left-swirling spirals turn into radial spokes in the following frame, and then right-swirling spirals in the next. Just like turning a radio dial, I think sieves gets tuned in one (or many) at a time until they show as straight spokes at maximum ‘tuning’ (and often many different sieves can be be seen in action at the same time in a frame):

Here’s an example frame from this program exhibiting these spokes (the frames immediately prior and after this display as spiral arms):

A Spoked 'Nixon' Spiral

So, what do you think? Is it the Sieve of Eratosthenes we’re seeing here. And, if so, does it have implications for understanding more about prime numbers, or even finding them more easily, for example?

  • Soloride

    It like you are tuning into a frequency. The prime number frequency. Reminds me of these videos: https://www.youtube.com/watch?v=YedgubRZva8

  • http://robinnixon.com/ Robin Nixon

    I think you are right – this does seem sort of similar.

  • Łukasz Majewski

    Hi Robin,

    I found your spiral about a week ago and found it intriguing and even implemented my own application to produce them (it plots other kinds of prime sequences too). Not sure whether you investigated this further but I also found this answer to the puzzle: http://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers

    Warm regards,
    Łukasz Majewski

  • http://robinnixon.com/ Robin Nixon

    Here’s something else I made a few months later – a fully interactive real-time animation that you can change many parameters with (try clicking inside the rotating display too :) http://primedancer.com