# The Doorways Problem and Sturmian Words

## TL;DR

This paper explores the geometric properties of lines passing through doorways in hallways, linking these configurations to Sturmian words, classifying line slopes, and establishing a complete metric space structure for Sturmian sequences.

## Contribution

It classifies slopes of lines passing through doorways in finite and infinite hallways and introduces a complete metric on Sturmian sequences, connecting geometric and combinatorial properties.

## Key findings

- Classified slopes of lines passing through doorways for finite and infinite hallways.
- Established a metric on configurations that preserves line-of-sight properties under limits.
- Proved the set of Sturmian sequences is complete under the induced metric.

## Abstract

The doorways problem considers adjacent parallel hallways of unit width each with a single doorway (aligned with integer lattice points) of unit width. It then asks, what are the properties of lines that pass through each doorway? Configurations of doorways closely correspond to Sturmian words, and so properties of these configurations may be lifted to properties of Sturmian words. This paper classifies the slopes of lines of sight, lines that pass through each doorway, for both the case of a finite number of parallel hallways and an infinite number and their consequences for Sturmian words. We then produce a metric on configurations with an infinite number of hallways that preserves the property of admitting a line of sight under limits. Pulling back this metric to $\mathbb R$, we produce the Baire metric under which the irrational numbers form a complete metric space. Pulling back this metric to the set of all Sturmian sequences, we show that the set of all Sturmian sequences is complete with this metric (unlike with the standard metric).