Packing $1.35\cdot 10^{11}$ rectangles into a unit square

TL;DR

This paper demonstrates that over 135 billion rectangles of decreasing sizes can be packed into a unit square, providing new bounds on the minimal area extension needed for such packings.

Contribution

The paper presents the first large-scale packing of over 135 billion rectangles into a unit square, improving bounds on the minimal epsilon for packing infinitely decreasing rectangles.

Findings

Successfully packed 1.35×10^{11} rectangles into a unit square.

Provided an estimate for the minimal epsilon in the packing problem.

Extended the known bounds for packing decreasing rectangles.

Abstract

It is known that $\sum\limits_{i=1}^{\infty} \frac{1}{i (i+1)} = 1$. In 1968, Meir and Moser asked for finding the smallest $\epsilon$ such that all the rectangles of sizes $1/i \times 1/(i + 1)$ for $i = 1, 2, \ldots$, can be packed into a unit square or a rectangle of area $1 + \epsilon$. In this paper, we show that we can pack the first $1.35\cdot10^{11}$ rectangles into the unit square and give an estimate for $\epsilon$ from this packing.