Mark
For the mathematics behind another very practical application, you could look at tiling also.
Require imaginary numbers? Not really. But describing AC current (or for that matter electromagnetism) succinctly requires either the use of complex numbers (sums of real numbers and imaginary numbers*) or something with the same algebraic structure.
You can dodge actually using them, if you really don’t like complex numbers, by using square matrices of real numbers whose diagonal entries are equal, and whose off-diaganal entries are negatives of each other. (Of course, under matrix addition and multiplication, these are just a copy of the complex numbers under their usual addition and multiplication.)
*I know *you* know this, but other readers of the thread might not.