Connections/motivations of "Sums of Two Squares" to/from other fields of math. I am to teach section 18 of "Elementary Number Theory" (Dudley) - Sums of Two Squares - to an undergraduate Number Theory class, and am having trouble cultivating anything other than a rote dissection of the lemmas/theorems presented in the text. 
The professor copies (exclusively) from the text onto the chalkboard during lectures, but I would like to present the students with something a little more interesting and that they cannot find in their text. 

What are the connections of the "Sums of Two Squares" to other fields of mathematics? Why would anyone care about solving $n = x^2 + y^2$ in the integers? 

I am aware of the norm of the Gaussian integers, and will probably mention something about how the identity $$(a^2 + b^2)(c^2 + d^2) = (ac -bd)^2 + (ad + bc)^2$$ is deeper than just the verification process of multiplying it out (e.g. I might introduce $\mathbb{Z}[i] $ and mention that "the norm is multiplicative"). 
What else is there? The book mentions (but only in passing) sums of three and four squares, Waring's Problem, and Goldbach's Conjecture. 

Also, I have seen Akhil's answer and the Fermat Christmas question, but these don't admit answers to my question.

Consider the Laplacian $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ acting on nice functions $f : \mathbb{R}^2 \to \mathbb{C}$ which are doubly periodic in the sense that $f(x, y) = f(x+1, y) = f(x, y+1)$. There is a nice set of eigenvectors one can write down given by
$$f_{a,b}(x, y) = e^{2 \pi i (ax + by)}, a, b \in \mathbb{Z}$$
with eigenvalues $-4 \pi^2 (a^2 + b^2)$, and these turn out to be all eigenvectors, so it is possible to expand a suitable class of such functions in terms of linear combinations of the above. 
Eigenvectors of the Laplacian are important because they can be used to construct solutions to the wave equation, the heat equation, and the Schrödinger equation. I'll restrict myself to talking about the wave equation: in that context, eigenvectors of the Laplacian give standing waves, and the corresponding eigenvalue tells you what the frequency of the standing wave is. So eigenvalues of the Laplacian on a space tell you about the "acoustics" of a space (here the torus $\mathbb{R}^2/\mathbb{Z}^2$). For more details, see the Wikipedia article on hearing the shape of a drum. A more general keyword here is spectral geometry. 
