Suppose you would like to cover the plane with congruent copies of Stars of David:
The goal is to overlap the least, to have a "thinnest" covering. One construction treats each Star of David as just an equilateral triangle, and ignores its other equilateral triangle, and then cover the plane with equilateral triangles. If my calculations are not in error, this leads to double-covering $3/9 = 1/3$ of the plane, something like this:
It seems unlikely this is the least double-covering fraction.
Q. What is the thinnest covering of the plane by congruent Stars of David?