The Pancake Theorem

Written by Borys Manikowski, 12/01/2024

Preface

Imagine you were making two pancakes, one for yourself, and one for your friend. Since you are an amazing cook, the first pancake you make is the best pancake you have ever seen, it was fluffy, golden and tender. You are a very inconsistent cook though, so the second pancake you make is the worst one you have ever seen, it was burnt, soggy and rubbery. Unfortunately, you ran out of ingredients to make another pancake, so you must somehow divide the ones you have made between you and your friend. Ideally, you’d like to keep things fair, so you want to cut both of them exactly in half, such that everyone gets half of the good one and half of the bad one. Since you are in a rush you only have time to make one cut, so you quickly put the pancakes on a plate and get ready to cut. Here a question arises: considering how you have made two pancakes, both having completely different shapes and sizes, and both placed randomly on a plate, are you always able to cut them both exactly in half using only a single cut? Turns out that yes, yes you are.

The situation described above is a real world example of the Pancake Theorem, the two-dimensional case of the Ham Sandwich Theorem. The Ham Sandwich Theorem states that for any collection of n objects in n-dimensional space, there exists a hyperplane that bisects all these objects. Thus, the Pancake Theorem roughly states that if you take any two 2-dimensional shapes you can always construct a straight line that divides both of them into two equal parts (in area). In this article I will present a proof for this theorem. Please keep in mind that proofs in combinatorial geometry and topology can get quite technical and complicated, yet I will try to provide an intuitive and interesting explanation.

Chapter 1: Defining a Pancake

As in any good proof, we must start by providing a formal definition of what we consider a pancake to be, in mathematical terms. Intuitively, a pancake is a region of a two dimensional plane which is finite in size, is one connected shape, and isn’t infinitely thin anywhere. More formally:

Let a pancake be a measurable set of points Ω in R2 such that:

1. ∀  p ∈ Ω, ∃ M>0 s.t. |p| < M 

2. ∀  p1, p2  ∈ Ω, ∃ a continuous path φ:[0,1] → Ω and where φ(0)=p1 and φ(1)=p2

3. ∀  p ∈ Ω, ∃ r>0 s.t. A ball with radius r centered around p, Br(p)∈ Ω

These three conditions ensure that the pancake fulfills our requirements. Condition 1, means that there exists a finite region which our shape can be enclosed in, meaning it is not infinite in size. Condition two ensures that from any point on the pancake you can get onto any other point on the pancake, meaning it is all connected (it is only one shape). Condition three states that for all points on the pancake you can construct a circle, no matter how small, such that the entire circle belongs to the pancake, meaning the pancake has a non-zero thickness everywhere. Finally, we don’t want the pancakes to overlap, to avoid any ambiguity regarding their areas. More formally if we have pancake one Ω1 and pancake two Ω2 we want to ensure that Ω1 ∩ Ω2 = ∅.

Chapter 2: Bisecting a Single Pancake

The first step in our journey of dividing the pancakes in half is to show that one pancake can always be divided in half by a straight line. To do this imagine a line going through the pancake going left to right (for this argument direction can be arbitrary, we are going left to right for simplicity). At first, the area on the left of the cut is zero and the area on the right is the area of the pancake. As the line moves the area on the left continuously increases, while on the right it continuously decreases, until all of the pancake is on the left of the line. Now, define a new function f(l) where l is the position of the line and f(l) is the fraction of the pancake’s area to the left of the line, thus we get:

• f(c)=0 means the entire pancake is to the right of the line

• f(c)=1 means the entire pancake is to the left of the line

• f(c) is continuous  (there are no “jumps” on the graph of f(c), a proof of this fact is left as an exercise to the reader)

By the Intermediate Value Theorem, there must be a point where f(c)=0.5, meaning the line bisects the pancake. As I stated before, this argument works regardless of the orientation of the line.

Chapter 3: Extrapolating to Two Pancakes

For the final part of the proof, let's try to extrapolate the previous results onto two pancakes. To complete this proof, we must show that there exists a line which cuts both pancakes exactly in half. To start we must define the direction of a line more formally. This can be done by using a unit vector n which represents a direction on S1, where S1 is a unit circle (that is S1={(x,y)R2:x2+y2=1}). Notice that each value of n will have one unique slope related to it, hence we can define a line l(n,t) to have a slope corresponding to n, where t is the position of the line along the normal vector n (how far away the line is from n). From what we have shown in Chapter 2, it is known that there exists a l1(n) which divides the first pancake in half. Similarly, there is a l2(n) which divides the second pancake in half. Now take l1(n)  and denote the area of the second pancake to one side of the line to be positive, and the area of the pancake on the other side of the line to be negative. You may arbitrarily choose which side is which, as long as you stay consistent. Now define a new function g:nR such that it is the sum of the two areas (remember you defined one area to be positive and one to be negative). Thus, we can conclude:

• g(n) is continuous

• g(-n)=-g(n) since by reversing the line, you are changing the positive and the negative area.

For the next part of the proof, we will have to use the Borsuk-Ulam theorem. The theorem states that for any continuous mapping of a n-sphere onto Rn there always exists a pair of anipodal points on the sphere that map to the same value. Intuitively, this can be explained with the example of earth, where the theorem guarantees that if you were to measure the temperature and the pressure everywhere on earth there must be two points exactly opposite of eachother where these two values will be equal. In the case of this proof the Borsuk-Ulam Theorem states that when a function g:S1R (that is, a function that takes in a point on a circle and outputs a real number, just like the function g) satisfies  g(-n)=-g(n), then there must be value n0 such that g(n0)=0. This means that there exists two points on the opposite sides of the circle, such that the output of g at those points is equal. This means that the area on one side of the line, minus the area on the other side of the line is zero thus, the areas are equal! I know using the Borsuk-Ulam Theorem is somewhat of a leap of faith, but unfortunately this is a vital part of the proof. I might write an article proving the Borsuk-Ulam Theorem in the future to give this more justification, but for now this must suffice.

Chapter 4: The End…

In conclusion, the pancake theorem shows us that for any two pancakes, we can always cut both of them in half with just one cut, no matter what their shapes, orientations, sizes, or positions on the plate are. This elegant proof really underlines the power of topology, and while this theorem does extend to higher dimensions (in fact it works for any number of dimensions), the two dimensional version is in my opinion one of the most interesting proofs in mathematics, while still being relatively simple and accessible.

Bibliography:

“Ham Sandwich Theorem.” Brilliant, https://brilliant.org/wiki/ham-sandwich-theorem/. Accessed 15 November 2024.

“The Pancake Theorem.” Ofir David, 1 July 2022, https://prove-me-wrong.com/2022/07/01/the-pancake-theorem/. Accessed 15 November 2024.