The ham sandwich theorem, demonstrated by mathematicians AH Stone and J. Tukey in 1942, tells us that if we want to share a sandwich with another person it will always be possible to do so with a single cut of the knife that divides, in one fell swoop, exactly in half each part of the sandwich. We explain the theorem in the three-dimensional case and try to understand why it works by analyzing a simplified two-dimensional version. Finally, we see a curious application of the theorem at a political level in the context of elections.
Explanation of the ham sandwich theorem
Let’s imagine having a sandwich made up of two slices of bread, perhaps irregular and different from each other, with a slice of ham in the middle placed there a bit haphazardly and let’s imagine that two diners want to divide the sandwich in the most equal way possible.
Dividing the sandwich approximately in half is easy, but things get complicated if each diner wants exactly half of the ham, half of the soft top of the sandwich and half of the crunchy bottom of the sandwich. Well, the ham sandwich theorem tells us that it is possible to do it, even with just one cut, but it tells us this using mathematical terms. Simplifying a little, and treating the slices of bread and the ham as 3 geometric figures, it can be expressed more or less like this:
Dates 3 figures in space, there is a plane that bisects (divides in half) exactly all three figures.
To better understand what it is we can look at the two-dimensional version of this theorem, also known as pancake theoremwhich has two flat figures as its protagonists (pancakes!) and according to which
2 geometric figures in the plane can each be divided exactly in half (into two parts with the same area) with a single straight line.
In this case we are talking about 2 figures, instead of 3, because the ham sandwich theorem refers to a number of figures not exceeding the number of dimensions considered: in space we are talking about 3 figures, in the plane we are talking about 2 figures.
In the figure below we see a simplified version in which one of the two pancakes it’s a circle. On the left side we see a straight line that divides the circle in half but which passes to the left of the second figure.
Now we slowly rotate the straight line around the center of the circle (as in the figure): we see how the straight line, which continues to divide the circle in half, begins to cut the second figure so that, at first, a small part of the figure is found to the left of the straight line while a larger part is found to the right of the straight line. Continuing with the rotation of the line, the portion of the figure located to the left of the line increases more and more, until, in the end, the entire figure is located to the left of the line (see right part of the figure above). The situation is now reversed compared to the beginning and during this process in which the straight line crosses the second figure there must necessarily be a moment in which it divides it exactly in half.
This is more or less the idea behind the theorem, but how do you make the right cut that divides the two pancakes, or the ham sandwich, exactly in half? Unfortunately, the theorem does not give us an answer, it is, in fact, an existence theorem which assures us that a solution to the problem exists but does not tell us what it is.
An application to the world of politics
In some states, when elections are held, the country is divided into districts and each district elects its own candidate, but the way in which the country is divided into districts can affect the outcome of the elections.
Let’s take an example, consider a state in which there are only 6 sympathizers of the green party and 2 sympathizers of the purple party, residing in different areas of the country. Suppose that the territory is divided into two districts, the one to the north and the one to the south (see figure below on the left), and that each district elects a candidate: in this case the North district unanimously elects a green candidate and the South district elects by majority, 2 against 1, a purple candidate. In total, one green and one purple representative are elected, as if the green and purple voters were equal in number.
However, looking at the total number of voters in the country we realize that the majority is clearly in favor of the green party (6 against 2): one wonders if there is a division into districts that reflects this situation. This is precisely where the pancake theorem comes into play, thanks to which we know that it is possible to divide the state in such a way that the supporters of the 2 parties are distributed equally between the two districts, half green in one district and half in the other, half purple in one district and half in the other. In our case (see figure above, on the right) it is enough to divide the state into East and West districts: in this case in both districts the green candidates would win the elections by 3 votes to 1 and the result of the elections, with two green candidates elected, would reflect the overall majority.
