implicity

the best packing of centrally symmetric shapes on a plane are regular hexagons or linear transformations thereof,
the best covering of the plane by centrally symmetric shapes is with regular hexagons or linear transformations thereof,
the best / worst ditto for unsymmetric shapes is with triangles and in general
if you have a property of a convex polygon and a polygon with a limiting value exists, it will usually be a triangle
if the convex polygon has to be centrally symmetric it's probably a hexagon

if you know this, you might be able to pass one of my courses.
which is more than i am yet confident of doing.