The Complex Squaring Function (Rectangular Coordinates)

We can look at the graph of the squaring function w = z2 as we did for the graph of the identity function . We can approach this by finding the real coordinates of the points on the graph, first expanding the square of x + yi algebraically and then separating the resulting complex number into its real and imaginary parts: z2 = (x + yi)2 = x2 + 2xyi + (yi)2 = (x2 - y2) + (2xy)i = u + vi. Thus we can insert into our four-dimensional grid the points (x,y,x2 - y2,2xy) where (x,y) runs over the points of the unit square. We can then observe various views of that graph as we rotate it and the hypercube in four-space.

[More on what the pictures below represent. Link to projections from R4? Surface graphs? Real and imaginary parts?]

Unfortunately we observe that the graph of the squaring function does not stay "inside" the hypercube! To see why this is so, observe that the corner (1,1) of the square is sent to (1,1,0,2), and the other three corners are similarly sent to points with coordinates of absolute value greater than 1. We can decide to live with this, realizing that the distortions are going to be even greater for higher powers of z. [Or...]