Problem: Compare stacking efficiency of balls in a large volume - rectangular arrays settled one atop another vs triangular arrays similarly settled. Latter gives more balls per unit area, but layer height is greater.
Let ball diameter = 1.
Case 1: Area per ball = 1. Height = .707
Case 2: Area per ball = .866 Height = (that of a tetrahedron with side = 1)
(2/3)^.5 = .707
No difference!
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