A Numeration System for Fibonacci-like Wang Shifts

TL;DR

This paper introduces a novel numeration system for integers and two-dimensional grids based on Fibonacci-like sequences, and presents a set of Wang tiles that can tile the plane deterministically using this system.

Contribution

It defines a new numeration framework for Fibonacci-like sequences and constructs a finite set of Wang tiles that tile the plane deterministically based on this system.

Findings

A numeration system for $ Z$ and $ Z^2$ using binary alphabet.

A set of 16 Wang tiles that tile the plane.

A deterministic automaton for tiling based on position representations.

Abstract

Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $\mathbb{Z}$ and $\mathbb{Z}^2$ based on the binary alphabet $\{0,1\}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane described by a deterministic finite automaton taking as input the representation of a position $(m,n)\in\mathbb{Z}^2$ and outputting a Wang tile.