The Ammann-Beenker tilings revisited

TL;DR

This paper explores a new family of tilings related to Ammann-Beenker tilings, derived from digitizing 2D planes in 4D space, and shows they can be characterized through an optimization problem.

Contribution

It introduces two new tiles forming a one-parameter family of tilings that generalize Ammann-Beenker tilings via a simple optimization framework.

Findings

Family of tilings includes Ammann-Beenker as a special case

Tilings are digitizations of 2D planes in 4D space

New characterization via optimization problem

Abstract

This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann-Beenker tilings as the solution of a simple optimization problem.