Covering Spaces and Packing Things
Let’s say you needed to ship different shaped items across the sea. If you had to put the items in a rectangular box, the most efficient shape to pack in the box would be a rectangle prism. There would be very little empty space with every box “nesting“ perfectly. The more and more sides that a prism has, however, the less and less efficient packing becomes. Finally, you are left with the sphere. If you have to put a certain amount of spheres into a box, how do you know the most effective way? Packing spheres is actually a highly studied mathematics topic and there are manly formulas for this exact problem.
If I threw a bunch of basketballs into one box, it would roughly only fill up 64% of the space. This somewhat makes sense as it is around 2/3 of the volume. It turns out that if you meticulously place spheres, you can average 74% of used volume. This 74% is an average found with different packing methods. If you fit the spheres into a box that perfectly fits the sum of the diameters, that would be row A. You can then either put spheres in between 3 spheres below them or directly above. The equation for the density of this packing is found by pi/3*sqrt(2) which yields about 74%. Numberphile provides a very good visual of how this equation came about, but I will briefly walk through it:
Start by stacking a simple four spheres on the bottom and one above them to create a square-based pyramid
Slice the top half of the top of the sphere off
Slice the sides of the bottom spheres off along with the bottom half of the bottom spheres
You are left with half of the top where and 1/4 of the four bottom spheres = one full sphere
This new shape is considered a “unit“ where a sphere with radius 1 would have a volume of 4pi/3
The entire box would have a radius of 4 sqrt(2) (2 x 2 x sqrt(2))
The density of the entire unit would be pi/3 sqrt(2) = 74.05%
Every different layout of packing spheres has a different name. The lowest possible density is around 5.5% but they average around 30-40%. What is interesting, is that no matter the base shape (square, triangle, hexagonal), the density will always come out to 74.05%. This is a little harder to visualize, but basically all of the pyramids, regardless of shape, can be broken into the same unit cube in the above example.
This problem, like most in topology, can be represented in other dimensions. In a two-dimensional world, the equivalent would be packing circles onto a plane. Mathematicians have considered hypersphere packing up until the 8th dimension, however, there is also packing known in the 24th dimension.
So that’s cool, but why is that useful. The most obvious reason is for, well, actually packing and shipping spherical products. The original drive behind this problem was finding an efficient way to pack cannonballs on a ship. The second has to do with chemicals and their makeup. Smaller spheres can fit in between the gaps created by the larger ones as long as their ratios are below 41% of the original sphere. Attoms also are found in this close packing makeup in crystals such as salt.
“Sphere packing“ takes a broader form in the world of data. Every piece of data sent can be thought of as different spheres that make up a giant message. Those spheres are transmitted and received, and if packed correctly, don’t alter that much. If there is any movement because the receiver knows what the shares should be a package like, the software can correct the message. This is a little bit abstract but contains the same principle of densely packing data. The dense this data is sent, the easier it is to correctly receive and the more efficient it is.
What I like about this sphere packing problem is that it is very easy to replicate and approximate yourself. Grab yourself a bag of marbles and a box, and with only a couple of experiments, you should find a density somewhere near the 65-74% mark. It is easy to replicate each time and easy to solve for the density by finding the volume of the cubes compared with the box.