The Symplectic Geometry of Penrose Rhombus Tilings

TL;DR

This paper explores the symplectic geometric structures underlying Penrose rhombus tilings, associating each tile with a singular symplectic space and revealing their invariance under specific Hamiltonian actions.

Contribution

It introduces a novel symplectic geometric framework for Penrose tilings, linking each rhombus to a unique singular symplectic space and analyzing their symplectic properties.

Findings

Each thick rhombus corresponds to a singular 4D symplectic space.

Each thin rhombus corresponds to a different singular 4D symplectic space.

The associated spaces are diffeomorphic but not symplectomorphic.

Abstract

The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4-dimensional compact symplectic space, while each thin rhombus can be associated to another such space; both spaces are invariant under the Hamiltonian action of a 2-dimensional quasitorus, and the images of the corresponding moment mappings give the rhombuses back. These two spaces are diffeomorphic but not symplectomorphic.