Knew It

A short note on how to interpret Fourier Series animations

When one searches for Fourier series animations online, these amazing gifs are what they stumble upon.

They are absolutely remarkable to look at. But what are the circles actually doing here?

Vector Addition

Your objective is to represent a square wave by combining many sine waves. As you know, the trajectory traced by a particle moving along a circle is a sinusoid:

This kind of looks like a square wave but we can do better by adding another harmonic.

We note that the position of the particle in the two harmonics can be represented as a vector that constantly changes with time like so:

And being vector quantities, instead of representing them separately, we can add them by the rules of vector addition and represent them a single entity i.e:

                                                  Source

The trajectory traced by the resultant of these vectors gives us our waveform. 

And as promised by the Fourier series, adding in more and more harmonics reduces the error in the waveform obtained.

        Have a good one!

**More amazing Fourier series gifs can be found here.

a lot of children - especially mentally ill children - end up traumatized not because someone was specifically hurting them but because their needs weren’t being met, or because their problems weren’t being seen, or because they were rendered particularly vulnerable by other aspects of their identity, like queerness or race. 

and it can be hard to look at your childhood and go “I was hurt” and also know that the hurt wasn’t deliberate. it’s uniquely painful to not have someone to blame. 

you do not have to excuse the people who hurt you, even if it was unintentional. & acknowledging your own pain does not necessarily entail blaming them for it.

you are allowed to do what you need to do in order to recover. 

There’s a freaking CHEETO in the White House and all you’re doing is blogging about sucking anime cock!

someone take me out. either in the date way or the assassination way

100% reblogging just for the comment

My favourite remix of this song