Laplace eigenvalues on regular polygons: A series in

For regular polygons P N inscribed in a circle, the eigenvalues of the Laplacian converge as N → ∞ to the known eigenvalues on a circle. We compute the leading terms of λ N / λ in a series in powers of 1 / N , by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The O ( 1 / N 2 ) term comes from Hadamardʼs formula, and reflects the change in area. This term disappears if we “transcribe” the polygon, scaling it to have the same area as the circle.