Aaron Aaronsonn Acquired an Acre of Abundant Apples

I keep a list of topics I'd like to use for these posts, and while this one is far from the oldest, it is from last fall: My friend Kevin came to visit and as a seasonal Michigan activity, he, Marika, and I went apple-picking. The orchard we went to sold bags of various sizes, and then would let you pick however many fit in the bag. Being physicists, Kevin and I began to wonder about the packing efficiency of the apples: Given the constant volume of the bag, is it better to pick more small apples, or fewer large ones?

For a given size, we want to get as many as possible into a space. This idea is usually represented with a lattice: an arrangement that repeats over some grid. Here's a pair of examples in 2D:

On the left is a square lattice, while the right is a hexagonal lattice. Our situation is 3D though, which makes things a little more complicated. If we approximate the apples as spheres, then we can use Gauss's solution for the density:

This is the density of apples inside the bag. There's a further complication though: The apples have cores that we'll be throwing away, so we have to consider the amount of each apple that the core takes up. I was surprised to find a paper studying exactly that, for the purpose of designing a mechanical coring device. They give the apple and core diameters for a few different varieties, and the relationship is roughly linear:

With the fit, we can relate the apple's diameter to its useable volume:

The total number of apples that fit in a bag is

where the brackets indicate rounding down to the nearest integer, since (hopefully) we're only picking whole apples. The total volume of useable apple then is

Below is a plot of the apple diameter vs the percentage of the bag wasted as air or core. The dotted lines represent the range of diameters from the study above, but I decided to extend the range to some more extreme values.

On both ends of the scale, our approximations begin to break down: On the left, the core shrinks faster than the useable apple, so we get down to the minimum 26% waste due to the packing inefficiency. On the right, you can't fit many 6-inch apples in a bag, so there are some sharp jumps. On the whole smaller apples are better, but in the realistic range, there isn't much difference. Clearly, further study is needed – Back to the orchard!