Ammann Bars for Octagonal Tilings

TL;DR

This paper introduces a general method for creating aperiodic octagonal tilings using Ammann bars, based on subperiods, and demonstrates it with a new set of decorated prototiles related to Cyrenaic tilings.

Contribution

It provides a novel, general construction approach for Ammann bars in octagonal tilings, expanding understanding of non-periodic tiling structures.

Findings

A new method for cut and project tilings using subperiods

Construction of 36 decorated prototiles for Cyrenaic tilings

Illustration of non-periodic tilings with Ammann bars

Abstract

Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tiles, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.