Weak local rules for planar octagonal tilings

TL;DR

This paper characterizes planar octagonal tilings that have weak local rules, revealing they are all based on quadratic irrationalities, confirming a long-standing conjecture from the 1990s.

Contribution

It provides an effective characterization of such tilings and proves they are based on quadratic irrationalities, confirming Thang Le's conjecture.

Findings

All planar octagonal tilings with weak local rules are based on quadratic irrationalities.

The paper offers an effective method to characterize these tilings.

It confirms a conjecture from the 1990s regarding their algebraic nature.

Abstract

We provide an effective characterization of the planar octagonal tilings which admit weak local rules. As a corollary, we show that they are all based on quadratic irrationalities, as conjectured by Thang Le in the 90s.