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The Math Trick Behind MP3s, JPEGs, and Homer Simpson's Face
Over a decade ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn't a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 and JPEG compression, X-ray crystallography, voice recognition, and PET or MRI scans. This mathematical tool—named the Fourier transform, after 18th-century French physicist and mathematician Joseph Fourier—was even used by James Watson and Francis Crick to decode the double helix structure of DNA from the X-ray patterns produced by Rosalind Franklin. (Crick was an expert in Fourier transforms, and joked about writing a paper called, "Fourier Transforms for birdwatchers," to explain the math to Watson, an avid birder.)
You probably use a descendant of Fourier's idea every day, whether you're playing an MP3, viewing an image on the web, asking Siri a question, or tuning in to a radio station. (Fourier, by the way, was no slacker. In addition to his work in theoretical physics and math, he was also the first to discover the greenhouse effect.)
So what was Fourier's discovery, and why is it useful?
The story provides great visual examples of how even complex waves can be approximated by a series of sine waves summed together. Further, the parameters to the sine waves and a much more concise description of the approximated item. Examples are given of a roughly-square wave. Another example uses circles instead of sine waves. A great YouTube video shows these in action.
Wish I had this available to me before I was taught FT and FFT in college!
(Score: 3, Informative) by Anonymous Coward on Tuesday June 11 2019, @03:28PM (2 children)
This is a great visualization: http://bgrawi.com/Fourier-Visualizations/ [bgrawi.com]
And if you want to roll up your sleeves and see another great visualization: https://www.youtube.com/watch?v=spUNpyF58BY [youtube.com]
(basically, that YT site is great for ANY kind of mathematical visualizations)
(Score: 4, Interesting) by AthanasiusKircher on Tuesday June 11 2019, @04:14PM (1 child)
Agreed. For those who haven't explored 3Blue1Brown's math channel [youtube.com] on YouTube, it has some of the best visualizations for a lot of lower-level college math concepts, combined with explanations that try to develop intuition of rather complicated math concepts (often by geometrical arguments and visualizations).
Digression: I love most of his videos on basic stuff, like intro calculus. My one very minor criticism is with more advanced topics, there is often an odd combination of handwaving through some really complicated ideas while simultaneously explaining really basic stuff unnecessarily. Take the Fourier Transform video parent linked, for example: 3Blue1Brown spends a bit of time trying to justify/explain wrapping a sine/cosine wave around the origin as some sort of visual metaphor. Except anyone with even a really, really basic sense of polar coordinates (and graphs of sine/cosine functions with varying frequencies in them) would know immediately where those graph shapes are coming from, why they end up creating a cardioid as the time interval of graphing lines up with the wave frequency, etc. Probably a 3-minute primer on polar coordinate graphing as part of that 21-minute video would have been more effective than some handwaving vagueness about wrapping waves around the origin and being mystified as the "center of mass" lines up when the waves are graphed in a certain way.
And yeah, the "center of mass" thing is then really handwaved over. We ignore basic polar coordinates, instead taking an excessive amount of time to explore properties that would be apparent to anyone who has spent 10 minutes graphing sines and cosines in polar coordinates, but then he introduces this "center of mass" thing with basically no explanation. What does "center of mass" mean for that wave? How is it calculated? Later, suddenly an integral appears, and then we're told that has something to do with center of mass... and again, a 30-second digression on how integration is used in physics to determine center of mass would have made it feel like an actual explanation. Same thing when Euler's formula stuff and complex coordinates are thrown in -- at least in that case he could point to a prior video he made on how that stuff works.
Again, these are relatively minor criticisms. I just am sometimes a little frustrated about how the author of that channel can be so good at explaining more basic concepts step-by-step, so you could actually show them to (say) an intro calc class, and they'd get a lot out of it. But the more advanced videos often skip over steps at precisely the most complex parts of the explanation. If I showed the Fourier video to a similar class, they'd wonder why he was wasting time because he doesn't assume knowledge of basic precalc concepts like polar coordinates and the complex plane, but then skips over several steps and suddenly throws in more advanced or borrowed concepts.
(Score: 2) by AthanasiusKircher on Tuesday June 11 2019, @04:18PM
Sorry -- all that said, it is truly an awesome video, with a much more detailed explanation of the math of Fourier transforms than TFA. So do watch it if you're interested in this stuff. (I didn't mean to be overly critical.)