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> It feels like the smaller marbles are denser but obviously they actually pack the same efficiency as the tennis balls That's not obvious to me. Won't the mar
by kens 6y ago
> It feels like the smaller marbles are denser but obviously they actually pack the same efficiency as the tennis balls
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%.
- dmurray 6y agoA "barrel" though is way larger than a tennis ball, so we're not close to the limiting case.
- Retric 6y agoPacking 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.
- OscarCunningham 6y agoAccording to http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html 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.
- OscarCunningham 6y agoIn 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).
- Retric 6y ago> 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.