Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of $\mathbb{F}_p$

TL;DR

This paper explores the construction of nonperiodic tilings derived from Stern's diatomic series, utilizing tiles decorated with elements of finite fields, revealing new additive and nonperiodic tiling patterns in one and two dimensions.

Contribution

It introduces a novel method for creating nonperiodic tilings based on algebraic structures related to Stern's diatomic series and finite fields.

Findings

New nonperiodic tilings in 1D and 2D

Tilings are additive in nature

Based on tiles decorated with elements of _p

Abstract

This paper presents the construction and various properties of substitution tilings related to Stern's diatomic series and based on tiles decorated with elements of $\mathbb{F}_p$ for some odd prime number $p$. These substitution tilings are additive in a sense that will be clarified later and lead to new nonperiodic tilings in one and two dimensions.