The eigenvectors of the Laplacian of a Ring graph with $n$ vertices are:
$x_k(u) = \sin(2\pi ku/n)$ and
$y_k(u) = \cos(2\pi ku/n)$
for $1\leq k \leq n/2$. The explanation according to Spielman's lecture notes is:
The best way to see that $x_k$ and $y_k$ are eigenvectors is to plot the graph on the circle using these vectors as coordinates. That they are eigenvectors is geometrically obvious.
Why is this "geometrically obvious"? I can see that embedding the graph into the plane using $x_k$ and $y_k$ as coordinates places the vertices uniformly over the unit circle. What I cannot understand is why the eigenvectors of the Laplacian must produce this embedding.