But what is a Fourier series? From heat flow to drawing with circles | DE4

3B1B is incredible. I loved his video ( ▶ this video ) using Fourier series to make epicycle animations. I worked through the details to make my own: [link]

in “Old blog of Matt Henderson, beautiful math animations” · View on HN

I wouldn’t call this 3blue1brown video a reference, but until I watched this (coming from an audio background), I didn’t quite grasp that a continuous stencil of any complex silhouette could be described in terms of a set of contributing vectors. Helped click that there is a perfect visual analogy to the deconstruction of complex audio to sine waves. ▶ this video

in “A ‘Useless’ Perspective That Transformed Mathematics” · View on HN

You can use epicycles to draw the actual orbits of planets. That's what "epicycles" are, or were: a fairly accurate model of the observed orbits of planets relative to Earth built through a composition of circular motions. One way to think about it is that you start by modeling the motion as a simple circular orbit. You then look at how the observed data differs from your circular model and add an epicycle whose parameters minimize the difference between your new model and your observations. Repeat this process until you get as close as you want to get. The success of heliocentric elliptic-orbit model isn't that it's able to represent something impossible with epicycles, but that it's dramatically simpler. Kepler's three laws are: 1) The orbit of a planet is an ellipse with the Sun at one of the two foci. 2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3) The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. In contrast, the epicycle models had 40-80 cycles with parameters tuned to match the observed data. I really like this video on how Fourier series work: ▶ this video

in “Is the great neutrino puzzle pointing to multiple missing particles?” · View on HN

> It remains a badge of honor for many who like to describe themselves as highly cultured or artistic to describe themselves as mathematically challenged, or to say that their brains aren’t wired for mathematics. Because many of those they hold in high esteem have made similar claims, there is no real social penalty to them for doing so. For many scientific illiteracy is in the same category as mathematical illiteracy -- but the problem starts there already. Everybody should at least know what a Fourier transformation does -- it's not necessary to exercise it rigorously, but at least an idea... I think this video is enough to give even laymans an idea: ▶ this video

in “Why Is Scientific Illiteracy So Acceptable?” · View on HN

Reminds of 'Drawing with circles' in 3b1b classic on Fourier series: ▶ this video So if we remove the condition of 13, everything is in fact made of circles only!

in “Animals Made from 13 Circles (2016)” · View on HN

IIUC: A real-parameter (r(theta) = sum(r_k cos(k theta))) Fourier series can only draw a "wiggly circle" figure with one point on each radial ray from the origin. A compex parameter (z(theta) = sum(e^(z_ theta))) can draw more squiggly figures (epicycles) -- the pen can backtrack as the drawing arm rotates, as each parameter can move a point somewhere on a small circle around the point computed from the previous parameter (and recursively). Obligatory 3B1B ▶ this video Since a complex parameter is 2 real parameters, we should compare the best 4-cosine curve to the best 2-complex-exponential curve.

in “Fitting an elephant with four non-zero parameters” · View on HN