# An Interactive Guide To The Fourier Transform

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. What does the Fourier Transform do? Here's the "math English" version of the above: The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the strength, offset, & rotation speed for every cycle that was found). Time for the equations? If all goes well, we'll have an aha! This isn't a force-march through the equations, it's the casual stroll I wish I had. From Smoothie to Recipe A math transformation is a change of perspective. The Fourier Transform changes our perspective from consumer to producer, turning What did I see? In other words: given a smoothie, let's find the recipe. Why? So... given a smoothie, how do we find the recipe? Well, imagine you had a few filters lying around: We can reverse-engineer the recipe by filtering each ingredient. Filters must be independent. See The World As Cycles Oh!

Differential Equations Explained $\cos$PLAY You're probably used to equations like $$(t-.5)(t-1)= 0,$$ where 'solving' means finding an unknown number. A differential equation (DE), by contrast, is a fact about the derivative of an unknown function, and 'solving' one means finding a function that fits. To visualize derivatives, we can draw a right triangle whose hypoteneuse is tangent to a function. If the triangle's width is $1$, then its height is the derivative. With that one weird trick, the plots to the right show how the derivative of $\sin(t)$ is $\cos(t)$. That's a pretty basic DE, though. Consider a cart rolling to a stop. The solution is a function $v(t)$ giving velocity at time $t$. It turns out the exponential function, $e^{-kt}$, has the properties  \begin{align} \frac{d}{dt}e^{-kt}=-ke^{-kt} && e^{-k\cdot 0}=1. To make the solution more intuitive, here you'll solve the cart's DE manually by picking a series of $\left( t, v \right)$ points. The first cart below obeys the $v(t)$ function you designed.