For squares, I think the answer is the only real positive root of the equation 1 + r = r^{2} + r^{4} + r^{5} + r^{7} , or r = .830+.
(The number in each square is the exponent of the side of the square.)

For triangles, in 2011, Maurizio Morandi improved my best packing. His best r is the positive root of the equation 1 + r = r^{2} + r^{3} + r^{5} + r^{9} , or r = .828+.

For hexagons, I think the answer is r = .807+.

Another interesting variant is to pack 1 polygon of side r^{1}, 2 polygons
of side r^{2}, 3 polygons of side r^{3}, and so on. In this case, the
largest r value for both the square and triangle is φ=(√(5)-1)/2, and there is no wasted space at all!

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