Contrary to what you might expect, this article is not actually about sausages. It is not even about food at all. Instead, the sausage catastrophe is a mathematical phenomenon that occurs when studying the theory of finite sphere packing.
Finite Sphere Packing
Imagine that you have a finite number of apples that you want to pack as tightly as possible in a plastic bag (I guess we’re talking about food again). Furthermore, for this thought experiment, we assume the apples are equally sized perfect spheres. In what arrangement would you pack these apples? Think about how you can use the least amount of plastic possible.
There are many possible ways to arrange your finite number of apples. These can be classified into three groups:
- Sausage packing;
- Pizza packing;
- Cluster packing.
The type of arrangement you choose determines the convex hull of the packing. This is basically the plastic bag you are putting your apples in. More formally, it is defined as the smallest convex set that includes all of the spheres (or apples). Below, I will quickly discuss each type of packing so that you have a better understanding of what each of these looks like.
Sausage packing is the arrangement in which all of the spheres lie in one straight line. Visualize all of your apples in a straight line, with your plastic bag around it. What does the shape of the plastic bag now look like? Exactly, like a sausage. So, the convex hull of this packing arrangement has the shape of a sausage, hence its name. I hope you can already see what the next packing arrangement will look like.
Another food-related packing arrangement is pizza packing. Here, all of the spheres lie in a single plane instead of a single straight line. If you have ever played pool or snooker, you will have seen this arrangement before. When racking the balls before the start of the game, the balls are arranged in a pizza packing.
If the spheres are not constrained to one single straight line or a single plane, but instead are arranged throughout 3D space, it is called cluster packing. This type of packing is most comparable to how apples are normally placed inside a plastic bag the way you find them in the supermarket.
Optimal Packing Leads to a Catastrophe
As econometricians, we are often most interested in how to optimize such an arrangement so that we waste the least amount of plastic. This is also good for the environment, of course. A logical and correct way to determine the optimal packing is to look at the packing density. If the packing is more dense, there is less empty space in the packing. This leads to a smaller convex hull for a certain amount of spheres.
The packing density is defined as the ratio between the volume of the spheres and the volume of the convex hull. Luckily, the late Hungarian mathematician László Tóth, among others, has done a lot of groundwork for us regarding optimal packing for a finite number of spheres. This is also where the sausage catastrophe has its roots.
It is quite easy to believe that for three or four spheres, the sausage packing is optimal. You just put your three or four apples in one straight line in order to reduce your plastic use. What might be less easy to believe is that this holds true for any number of spheres up to 55. So, even if you for some reason have 55 apples that you need to pack optimally, it is best to do so by putting them in one straight line. Whether this is optimal for transportation is a story for another time. Also, the same holds true for n = 57, 58, 63, 64. For all other finite numbers, the sausage packing is not optimal. Instead, there exists a cluster packing with a higher packing density.
The sudden shift from sausage packing to cluster packing was coined jokingly as the sausage catastrophe. The orderly, neat sausage packing is suddenly no longer optimal and is instead replaced by the not-so-simple cluster packing. As of yet, there is no clear explanation as to why this phenomenon occurs.
Moving Into Other Dimensions
It turns out that once we move away from three-dimensional space, the optimal packing becomes even more interesting. The sausage catastrophe still occurs in four-dimensional space. However, instead of occurring at n = 56, the transition from sausages to clusters is conjectured to happen only at around 377,000 spheres. That’s quite a lot of four-dimensional apples.
For the pizza lovers among us, I have less fortunate news. For all dimensions up to and including 10, it turns out that the pizza packing is never optimal. Instead, sausage packing or cluster packing always has a higher packing density, no matter the number of spheres. There is still hope, however: it is an open problem whether this holds true for all dimensions. So, maybe there exists a dimension in which the pizza packing comes out on top.
Sausage lovers can be pleased: Tóth’s sausage conjecture states that the sausage catastrophe no longer occurs when we go above the fourth dimension. Although this has not been proven completely yet, we know that for dimensions 42 and above this holds true. So, for your 42-dimensional apples, no matter how many of them you have, they are always optimally arranged in a single straight line.
Going to the Supermarket
The next time you go to the supermarket, you will now be aware of the inefficiency of the packing of the apples you buy. They should be arranged in the form of a sausage, and not as they are currently! Unless you buy over 55 apples of course. In that case, your apples are probably perfectly packed and you can go home knowing you’ve not wasted any plastic.