The ham sandwich theorem, demonstrated by the 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 which divides, in one fell swoop, exactly halfway through 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, let’s look at a curious one application of the level theorem political in the context of elections.
Explanation of the ham sandwich theorem
Let’s imagine we have a sandwich consisting of two slices of breadPerhaps irregular and different from each other, with one slice of ham in the middle it is placed there a little haphazardly and we imagine that two diners want to divide the sandwich as equally as possible.
Dividing the sandwich approximately in half is easy but things get complicated if each of the two guests wants to exactly half the hamhalf of the soft top of the bun and half of the crunchy bottom of the bun. Well, the ham sandwich theorem he tells us that it is possible to do it, even with a single cut, but he 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 one straight line that divides the circle in half but which passes to the left of the second figure.
Now let’s do it 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, start cutting the second figure so that, at first, a small part of the figure is to the left of the straight line while a larger part is 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 passes through the second figure there must necessarily be a moment in which he 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? The theorem unfortunately does not give us an answer and is, in fact, a existence theorem that there ensures that a solution to the problem exists but it doesn’t tell us what it is.
An application to the world of politics
In some states, when the elections, the country is divided into districts and each district elects its own candidate, but the way the country is divided into districts can affect the outcome of elections.
Let’s take an example, consider a state in which there are only 6 supporters of the Green Party And 2 sympathizers of the purple partyresiding 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 a manner That the sympathizers 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.
