Trig Waves and Divisibility

Finally uncovered some old matlab/octave generated graphs from my prime number research (2008).  I don't want to take the time to fully detail each formula/function, but the images are interesting to study, and you can probably figure them out if you really want to.

This first graph is the one that guitar harmonics reminded me of in an earlier post...  It's the absolute value of sine/cosine waves increasing in period and amplitude by integer amounts.

As you can see, taking the absolute value makes it much easier to manage, because you can cut the period in half, and you don't have to mess with those annoying negative numbers.

This graph is increasing sine waves, with the asymptotes showing how the trend would continue if more and more waves were plotted. In this graph, N=3.

You can see that the waves all equal zero whenever they reach a number that is divisible by each waves period.  More waves coincide at integers that are highly divisible, and an obvious pattern emerges, and repeats.

Note how the asymptotes alternate +/- (positive/negative).

This next graph is essentially the same function, but this time, N=50.  Although this looks much more chaotic, the same symmtry is still underlying. 

(the first three asymptotes are identical to the graph above... note how much more they've "filled-in" with the resulting waves...)

I haven't decided which I like better, sine or cosine - they each have their advantages when dealing with the prime numbers and divisibility.  Taking the absolute value of the waves chosen makes all of the asymptotes positive, as you can see in this graph:

This graph shows the exact same information as before, but now we are restricted to positive numbers.

I only used the increasing amplitude in my graphs to make the the symmetry easier to see.  Modifying the formula to give each wave equal amplitude results in this representation of the symmetry:

It's tougher to discern where the waves of each period start and stop.  I've filled in black dots for all of the prime numbers up to 60, and included the circles to show the symmetry of the spacing of the waves.  Remember, when the waves all coincide at the axis, the number is composite (divisible by other numbers - not prime).  This is a visual representation of the Seive of Eratosthenes made much quicker with symmetric waves.  Wherever the waves do not touch (up to a limit) will be the prime numbers.

So an easy way to see where the waves all line up is to take the SUM of the values of each wave at each integer.  Using the equal amplitude waves and taking the SUM results in this crazy graph:

This is the sum of equal amplitude cosine waves up to 50 partial sums (kind of reminds me of the stock market, but that's another post for later). The points in the graph where it seems to dip or have peaks and valleys is just another representation of the prime number symmtry.  Each major dip happens at a highly divisible number (e.g. 24, 30, 36, 48, 60).  A cool thing happens when you take sums of symmetric functions:  the sums also retain a similar symmetry.

I guess that's enough on this topic for now.  All of these graphs and functions are from research I did in 2008, so I've got plenty more to cover.  I'll bring on another installment soon enough!