## Thursday, 26 January 2012

### Self-similarity in composite patterns (expressed as difference of two squares)

Here's a minor example of self-similarity in the patterns I have been looking at.

The first row of each chart is N^2 - 1.

When we sieve for composites next to multiples of 24, the pattern is 6, 24 , 54, 96... (as seen in the first line of the diagram below).

For composites of 48, we halve this and get 3, 12, 27, 48... This is a series which ascends by 6X+3 at each step, with X increasing by 1 at each step.

For composites next to multiples of 96, the pattern is 6, 24 , 54, 96

For composites next to multiples of 192 it is once again 3, 12, 27, 48

Thereafter these two patterns are completely self-replicating, but increasingly stretched out, as can be seen below.

It's a bit fiddly to explain the reason why though it basically comes down to the fact that for each even number X in the pattern we find the even number 4X 4 spaces away . As a result, the pattern keeps repeating.

I won't get bogged down trying to explain this more formally as it doesn't really impinge on anything else I want to say. It is just interesting to note that in at least one aspect this pattern is perfectly self-replicating.

There are however more interesting things to be seen with respect to self-similarity in other parts of the pattern, which I will come to next.

Edit: Coming back to this post later on, I see that there is a simple way to explain what I was fumbling to say above in terms of differences.

For the series 3, 12, 27, 48 the first difference is 3, 9, 15, 21, and the second difference is 6. (eg the first difference increases by 6 each time).

For the series 6, 24, 54, 96, the first difference is  6, 18, 30, 42 and the second difference is 12 (so they are both twice the series above). when we halve this series we get the first, when we halve the first we get the second and lose the odd numbers, so this is an endless loop.