The Fermat variety of dimension r and degree m is a subvariety of CP(r+1) (the complex projective space of dimension r+1) defined by

For r = 1, we have Fermat curves. The equation above has three terms and the geometry of the Fermat curve is intimately linked with Fermat's Last theorem.

The Fermat varieties for m = r + 2 provide interesting examples of Calabi Yau manifolds. r = 1, m = 3 yields a two-dimensional torus, while r = 2, m = 4 is a K3 surface and, r = 3, m = 5 is a quintic threefold. Calabi-Yau's most striking feature is that they solve the Einstein vaccuum equations, making them of considerable interest in string theory.

Andrew Hanson found an interesting way to picture Fermat curves, and his technique seems to underly all of the popular representations of Calabi-Yau manifolds. It goes as follows. (See his paper, and see below for the url of his slides, which Behance won't accept as a valid link.)

First, because our picture will be 2 or 3-dimensional, let us look at a slice of the Fermat variety in CP2 by setting x_i = 0 for i > 2. The slice is then a Fermat curve.

Second, let's pick a patch in CP2 on which x0 is non-zero, which allows to set it to a convenient value and reduce the equations to x1^m + x2^m = 1, where x1 and x2 are complex. Hanson describes a clever way to parametrize the solutions to this equation.

Third, we see this as a system of equations for a real surface in R4, and project either to R2 or R3 to get an actual picture / 3d object.

In this project, we use Hanson's technique to plot the Fermat curves for m = 2, 3, 4, 5, using a random projection from R4 to R2. The images coming in paris are stereographic: look at the left image with your right eye and vice versa to see the Fermat curve in three dimensions.

URL for Hanson's slides: https://icerm.brown.edu/materials/Slides/sp-f19-w1/Illustrating_String_Theory_Using_Fermat_Surfaces_]_Andrew_Hanson,_Indiana_University,_Bloomington.pdf