Henry Baker <hbaker1@pipeline.com>

I'm curious as to what others think about this Fourier Transform video (1/26/2018). http://www.3blue1brown.com/ https://www.youtube.com/c/3blue1brown

Interesting. I've been playing with Fourier transforms for decades, but I had never thought of them that way before. Another visual representation is the waterfall display, in which time is the vertical axis, frequency is on the horizontal axis, and power is represented by brightness. In this representation, a Fourier transform can be represented by a 90-degree rotation. That makes it clear why doing a Fourier transform twice results in a time reversal. (It also results in a frequency reversal, but since negative frequencies are equivalent to positive, this makes no difference.) That representation also raises the question of what the math is behind smaller rotations. For instance what's the name of, and formula for, the 45 degree rotation, i.e. the transform that gives the Fourier transform if done twice. But mostly I'm not very visually oriented. I often get insights that make no sense to visual thinkers. For instance when an engineer friend tried to explain Laplace transforms to me, I was puzzled for a while, until I suddenly realized that he was talking about the Fourier transform of imaginary frequencies. Or, equivalently, of imaginary times. He was rather disturbed by that. But he later confirmed that the math comes out right -- which made him even more disturbed. Years later, I used a Fourier or Laplace transform of both real and imaginary frequencies to predict who would attend the next meeting of a science fiction club for which I had decades of attendance data. (As club secretary, I prepared the single-page sign-in sheet, and I wanted to list the names of those likely to show up, and leave blanks for others to fill in with their names.) It worked quite well. For instance compare the attendance of the April 1st meeting that I printed in the club journal distributed at that meeting as a prank to the actual attendance of that meeting that I printed in the following month's journal, <http://wsfa.org/journal/j05/4/#a1m> and <http://wsfa.org/journal/j05/5/#ffm>.