Parallel packing equilateral triangles into a square

TL;DR

This paper proves that a collection of homothetic equilateral triangles with total area up to c4d7c4d/4 can be packed into a unit square, establishing the tightness of this area bound.

Contribution

It establishes a tight area bound for packing homothetic equilateral triangles into a square with parallel sides.

Findings

Any such collection with total area c4d7c4d/4 can be packed into the square.

The bound c4dd/4 is proven to be tight.

The packing result applies to triangles homothetic to a fixed equilateral triangle.

Abstract

Suppose that $I$ is a unit square and that $\Delta$ is an equilateral triangle with a side parallel to a side of $I$. In this note, we prove that any collection of triangles homothetic to $\Delta$, whose total area does not exceed $\frac{\sqrt{3}}{4}$, can be parallel packed into $I$. The upper bound of $\frac{\sqrt{3}}{4}$ is tight.