# Lattice approach to plane colorings

## TL;DR

This paper introduces a lattice spin model as a physical analogy for plane coloring problems, providing insights into the minimum number of colors needed for proper plane coloring.

## Contribution

It presents a novel physical model approach to the plane coloring problem, offering approximate solutions and bounds on the minimum number of colors required.

## Key findings

- Minimum energy configurations suggest at least seven colors are needed.
- Approximate lattice colorings with 2-7 colors support the seven-color conjecture.
- The model provides a new perspective on the chromatic number of the plane.

## Abstract

I propose a fixed-range interaction multicomponent spin model, to be used as a physical analog to problems in plane geometry. Specifically, the model is applied to the open problem of the chromatic number of the plane. When spin values are interpreted as colors, the lowest energy configurations of the lattice spin system can be interpreted as approximations to plane colorings. In general minimum energy configurations of the model give optimal colorings, corresponding to minimum probability of any color realizing distance one. Approximate optimal lattice colorings with two to seven colors towards the continuum limit suggest that a true coloring of the plane cannot be achieved with less than seven colors.