Lines lines and more lines
The Exclusion lines in the first three Ulam spiral iterations produce very similar patterns.
I noticed three things when I look at them together
- The diagonal Exclusion lines rotate through each iteration
- The third Ulam has a different pattern of diagonal Exclusion lines
- The vertical and horizontal Exclusion lines are the same distance apart in each iteration
The diagonal lines rotate between each iteration because of the way the integer number spiral distributes the integers for odd and even increments.
The third Ulam spiral has a different pattern of diagonal exclusion lines because I deemed the 'missing' lines (every third one) as redundant when analysing the six layers of exclusion lines I came up with whilst undergoing image analyis of the third Ulam iteration. By that stage I was overlaying patterns of 'possible primes' to help determine which Exclusion lines actually contributed to knocking out integers that could have been prime. I think that because I had learnt that primes must be of the format 6n +/- 1 I did not include them in the over lay of possible primes.
The observation that the vertical and horizontal Exclusion lines are the same distance apart took me onto the next level.
Whilst visualising only the vertical and horizontal lines I imagined the Ulam spiral going through an inifinte number of iterations. It seemed to me that the only numbers that are prime are those that can survive an inifite rotatation through the Ulam spiral whilst not landing on these vertical and horizontal exclusion lines that remain the same distance apart through all Iterations. Almost like a fixed grid of wires that the integer spiral is spun through... a seive of sorts.
This is when I decided to work on my software so I could layer the Ulam spiral iterations.