Ammann Tilings in Symplectic Geometry

TL;DR

This paper explores Ammann tilings within symplectic geometry, constructing singular symplectic spaces linked to the tilings, revealing their diffeomorphic but non-symplectomorphic nature and symmetry properties.

Contribution

It introduces a novel association between Ammann tilings and explicitly constructed singular symplectic spaces, highlighting their geometric and symmetry characteristics.

Findings

The constructed symplectic spaces are diffeomorphic.

They are not symplectomorphic.

The spaces inherit symmetries from Ammann tilings.

Abstract

In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic but not symplectomorphic. These spaces inherit from the tiling its very interesting symmetries.