Analysis of a Fivefold Symmetric Superposition of Plane Waves

TL;DR

This paper demonstrates that a symmetric superposition of five plane waves can be expressed as a convergent series, and explains how its extrema relate to Penrose tilings, linking wave superpositions to quasicrystal patterns.

Contribution

It introduces a novel series expansion for fivefold symmetric plane wave superpositions and connects their extrema to Penrose tiling vertices.

Findings

Series converges pointwise and uniformly in disks in R^2.

Extrema locations approximate Penrose tiling vertices.

Provides heuristic explanation for extrema distribution.

Abstract

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.