Geometric Series of Polygons

What is the largest r for which polygons of sides 1, r, r2, r3, . . . can be packed inside the same polygon of side 1+r ?

For squares, I think the answer is the only real positive root of the equation 1 + r = r2 + r4 + r5 + r7 , 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 = r2 + r3 + r5 + r9 , or r = .828+.

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

Another interesting variant is to pack 1 polygon of side r1, 2 polygons of side r2, 3 polygons of side r3, 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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