Parallel packing a square with isosceles right triangles and equilateral triangles

Abstract

Suppose that $I$ is a unit square. Let $T$ (resp. $\hat{I}”$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\hat{I}”$), whose total area does not exceed $\frac{1}{2}$ (resp. $\frac{\sqrt{3}}{4}$), can be parallel packed into $I$. These upper bounds are tight.