T-Tetrominos in Arithmetic Progression

TL;DR

This paper explores the inherent order in tilings of rectangles by T-tetrominos, focusing on the length of arithmetic progressions of tiles, extending known tiling conditions with a new measure of structural regularity.

Contribution

It introduces a novel approach to analyze the order in T-tetromino tilings using arithmetic progressions, beyond existing tiling existence results.

Findings

T-tetromino tilings exhibit minimal arithmetic progression lengths under certain conditions.

Regular, periodic tilings correspond to long arithmetic progressions.

The study quantifies the degree of order necessary in tilings of rectangles by T-tetrominos.

Abstract

A famous result of D. Walkup is that an $m\times n$ rectangle may be tiled by T-tetrominos if and only if both $m$ and $n$ are multiples of 4. The "if" portion may be proved by tiling a $4\times 4$ block, and then copying that block to fill the rectangle; but, this leads to regular, periodic tilings. In this paper we investigate how much "order" must be present in every tiling of a rectangle by T-tetrominos, where we measure order by length of arithmetic progressions of tiles.