What do Polygons and Galois Theory Have in Common?

Galois Theory is all about symmetry. So, perhaps not surprisingly, symmetries found among the roots of polynomials (via Galois theory) are closely related to symmetries of polygons in the plane (via geometry). In fact, the two are highly analogous! The book Galois Theory (2ed, p. 59) by Joseph Rotman unfolds this analogy for us, as the picture below illustrates.

(Not sure what Galois Theory is? Find out here!)