Analysis of a Fivefold Symmetric Superposition of Plane Waves

We show that a symmetric superposition of five standing plane waves can be expressed as an infinite series of terms of decreasing wavenumber, where each term is a product of five plane waves. We show that this series converges pointwise in R^2 and uniformly in any disk domain in R^2. Using this series, we provide a heuristic argument for why the locations of the local extrema of a symmetric superposition of five standing plane waves can be approximated by the vertices of a Penrose tiling.