Comments by "vangrails" (@vangrails) on "3Blue1Brown" channel.

  1. New answer. The amplitudes are calculated by calculating integrals and the integral of a function doesn't that much if you change the function a little. Therefore the amplitudes don't change that much if you change the drawing a bit.
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  2. Yes, the arrows always keep the same length and the same angular velocity and this angular velocity is always a multiple of the angular velocity of the second arrow. The first arrow doesn't move, it is a constant.
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  3. This helping happens when all the arrows have more or less the same phase. So maybe you can call it phase uniformity?
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  4. The convergence of the Fourier series is a hard question: https://en.wikipedia.org/wiki/Convergence_of_Fourier_series
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  5. Take f(x) = sin x + 1/100 * sin 2x + sin 3x. Then the second amplitude is much smaller than the first and the third.
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  6. How to approximate a square wave by summing sine functions? A square wave has 90 degrees corners. Sine functions are smooth and don't have corners. You can only get a square wave out of summing sine functions by summing an infinite amount of sine functions and that means that you have to do an infinite amount of calculations on your computer. If you use a finite amount of sine functions then you get imprecise corners.
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  7. Sheet music is a compression of music, you only write the frequency and the duration of that frequency. That way Beethoven could compress music into a format that could be stored on paper and skilled people can still decompress what he has compressed centuries ago. It is of course not a lossless way of compressing music but that also has its charm. Every musician can choose how to fill the gaps.
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  8. You can consider it to be the center of mass.
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  9. I say Yes.
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