A Frameless 2-Coloring of the Plane Lattice

TL;DR

This paper introduces a simple 2-coloring method for the plane lattice that ensures no picture frames exist, contributing to the understanding of lattice colorings with specific structural constraints.

Contribution

It presents a novel, straightforward 2-coloring scheme for the plane lattice that guarantees the absence of picture frames, a new structural property in lattice colorings.

Findings

The 2-coloring scheme successfully eliminates all picture frames.

The method is simple and easily applicable to infinite lattices.

This work advances the theory of lattice colorings with structural restrictions.

Abstract

A picture frame in two dimensions is a rectangular array of symbols, with at least two rows and columns, where the first and last rows are identical, and the first and last columns are identical. If a coloring of the plane lattice has no picture frames, we call it frameless. In this note we show how to create a simple 2-coloring of the plane lattice that is frameless.