Have you ever tried to visualise the graph of a complex function $ f:\mathbb{C}\to\mathbb{C}$? The problem with complex functions is that usually we graph a complex number as an ordered pair $ (x,y)$ on a Euclidean plane, which corresponds to $ z = x + iy$. Unfortunately, this means that if we want to graph complex functions as we do real functions, we need to draw the graph in four-dimensional space! Some people have actually claimed the ability to visualise this, but I do not!

However, if we use time as a dimension, we could represent four dimensions as a moving three-dimensional image in time, like a movie. Sometimes, it’s hard to draw three dimensions in two dimensions, though we don’t actually lose too much because we can only see a two-dimensional picture of a three-dimensional scene at any given time.

There are a few animations in this post; they may take a few seconds to load, or a few minutes on a slower connection.

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