Explained What is a Fourier Series? (by drawing circles)

Yes, that's because a sinusoid is a circular function.

Here's an exercise: Take a Slinky outside in the bright sun. Hold it so it's perpendicular to the sun's rays and stretch it out. Now look at the shadow... it's a sinusoid.

Why is a sinusoid classified as a circular function? Because a sinusoid is a circle, spread out over space and time.

So when you're looking at a sine wave on your oscilloscope, you're really looking at the shadow of a circle! Unfortunately, we don't have 3-D scopes... they don't exist. But if they did, it'd make it much easier to design and troubleshoot circuits. I guess a scope's Lissajous Curve function comes close, but it's still not 3-D.

You'll note a circle's circumference is equal to 2π radians, and a full wavelength of a sinusoid is also equal to 2π radians.

What is a "radian"? Well, take a piece of string and measure the radius of any given circle. That's the radian length for that circle. If you then lay that piece of string along the line forming the perimeter of that circle, you'll find it takes 6.2831853 string lengths to go fully around the circle. So a "radian" is another word for the radius, geometrically transformed so it lays along the perimeter of the circle. It takes 2π of these radians to go fully around the circumference of the circle. C=2πr.

You'll also note the circle's diameter is equivalent to the sinusoid's peak-to-peak wave height, and its radius is equivalent to the peak wave height.

This goes right down to the quantum mechanical level... think of a bound electron in orbit about its nucleus. The electron must have an integer number of de Broglie wavelengths in its orbit, otherwise a destructive orbit is established which causes the electron to 'spiral inward' until an integer number of de Broglie wavelengths make up its orbital path length.

In other words, the electron must have a full circle or circles in its orbit, or the orbit has destructive interference which lowers orbital energy level. Partial wavelengths (partial circles) are not allowed.

Why is this? Because the electron orbital is a standing wave.

This is the reason why there are discrete electron shells (energy states). These electron shells obey Newton's Second Law.

Now, that comes with a caveat... as an atom becomes more electronically complex, the electron shell model becomes less exact. There aren't really electrons in neat little 1s, 2s, etc. shells, there are electron wavefunctions (standing waves) which resonantly interact to form a single multi-electron wavefunction as the electrons orbit the nucleus, and which we can approximate using the electron shell model.

And that translates up to chemistry... molecules are made up of resonant-frequency atoms, and if the resonances of the atoms making up that molecule are not in resonance with each other, the molecule is less stable. On the same note, if you have an otherwise-stable molecule and add an atom or molecule which disrupts its resonance, that otherwise-stable molecule can become explosively unstable. We'd empirically figured all this out long ago, but it wasn't until just recently that we figured out why it works the way it does, and with an ever-expanding knowledge base bolstered by computer modeling of atoms and molecules, we can form unique molecules which allow us to do things we otherwise wouldn't be able to do.

It's only a matter of time before we discover how to 'unwind' the resonant frequencies of matter directly back into energy... not fusion, not fission... matter degeneration. It would emit a lot of energy... 1 gram of matter (the approximate weight of a US dollar bill) would power a 100 W lightbulb for ~30,000 years or power your 100 HP car engine for ~35 years.

I have an idea of how to do so based upon my knowledge of quantum mechanics, but introducing that to a world which is hell-bent on building weapons of mass destruction and subjugating people would seem to be a bit more than a bit irresponsible. A 10 kg device built as a quantum bomb would make the simultaneous detonation of all the world's nuclear weapons seem like a firecracker by comparison.