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!

quaternions - hamilton

       

Stumbled upon this image... pretty cool.  Description below is roughly from Wikipedia:

Quaternions are a number system used to extend the “complex numbers”. Hamilton defined a quaternion as the quotient of two “directed lines” in a three-dimensional spaceor equivalently as the quotient of two vectors.Quaternions can also be represented as the sum of a scalar and a vector.
Directly applies to mechanics in 3-dimensional space - (physical existence as we know it)

Guitar Harmonics and Math/Physics

http://en.wikipedia.org/wiki/Guitar_harmonics

Found this nice layout of the harmonics on a guitar fretboard, and noticed it is identical to the wave-forms I've been using to try to work on the Goldbach Conjecture and other problems with prime numbers/divisibility.

I find it amazing that the physical phenomenon that we hear as music is essentially just waves of tones/frequencies that directly relate to the divisibility of integers.  I've always liked the quote:  "Music is the pleasure the human mind experiences from counting without being aware that it is counting." (Gottfried Leibniz).  With this layout of the harmonics, you can easily see the symmetry that arises when multiple waves are overlapped. 
Why does that matter?  Well, I guess it depends on your perspective of our universe... is it infinitely divisible (yielding smaller and smaller particles as you get smaller and smaller, never reaching a "smallest" particle) or is the universe finite (if so there would have to be some "smallest" particle, from which all other matter is created)? 

If finite, then each larger and large particle could possibly be defined in terms of these "smallest" particles.  However, there could be some order to the crazy quantum mess that we currently understand to be random.  Look under the red domes in the picture above... when all of the waves are overlapped, and their wave-lengths are not coinciding, it looks pretty chaotic, but once you zoom out, you can see that the apparent chaos is actually just a small portion of a fully symmetric structure.

If the universe is truly infinite, and infinitely divisible, then questions of continuity and differentiability come into play.  Most of our physics is contingent upon equations involving limits and infinite sums, which would require this infinite divisibility.  But we can't ever really know at the smallest or largest scales how "accurate" our math/science is. 

I believe the same symmetries that exist in music are also present in mathematics (discrete and continuous), and that it is those symmetries which we have detected as reproducible patterns in our physical world (yielding functions and formulas based on our observations).  We may not have the exact formulas or functions to yield perfectly accurate results, but at the same rate, our approximations of objects like spheres and circles would never come close to the "real" shape, which can't exist in our universe as we understand it - but even though the shape can't physically exist, we can figure out an exact formula for that shape to base our approximations on. 

It is the same with the laws of physics.  Our formulas and functions are just trying to explain the perfect shape that we see should be there, but that we may never acheive due to our finite limitations.

I wonder if Music Theory has ever been directly applied to quantum mechanics and particle physics?  Sounds like a blog for another day...  will do some research and see!