Packing efficiency is usually defined for an infinite space because finite containers don’t always favor the same size object.
Consider a marble with a diameter of 2inch and a tennis ball with diameter if 3 inches. A cube of length 9.01 inches now favors the tennis balls where one of 8.01 inches favors the marbles.
Re:2+1=4, by quinoa
by OscarCunningham on Saturday July 04, 2020 @12:47AM
(#23730185)
According to http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html a cube of side 9.01 can fit 27 balls of diameter 3, but 100 balls of diameter 2. Since 27×3^3 = 729 and 8×2^3 = 800, the balls of diameter 2 are more efficient in this case.
Re:2+1=4, by quinoa
by OscarCunningham on Saturday July 04, 2020 @04:39AM
(#23731026)
In fact it seems that the diameter 3 balls are most efficient only for cubes of edge-length between 3 and 2+sqrt(2) = 3.414.... These are the sizes for which you can fit in 1 diameter 3 ball, but can't fit in 4 diameter 2 balls. Above that, diameter 2 balls are more efficient (except that a cube of edge-length 6 can fit in the same volume either way; 27 diameter 2 balls or 8 diameter 3 balls).
> Since 27×3^3 = 729 and 8×2^3 = 800
I think you mean 100 x 2^3 = 800. But, yer correct that’s what I get from using a half remembered example without checking.
That's not obvious to me. Won't the marbles pack more efficiently due to the edges of the container? As a limiting case, consider a container slightly smaller than 1 tennis ball: the packing efficiency of tennis balls will be 0%, while marbles will be something like 60-70%.