Favorite Theorem #1

The Pancake Theorem

Let me begin by saying that these theorems that will follow in this section of my blog will not be in any particular order, especially not in order of importance or preference. They are just popping up as I recall them or come across them.

Today’s Theorem, the pancake theorem, is one that I just came across today. I don’t remember if I have ever seen it before, but I have nominated it as my Theorem du jour.

The theorem says that the area of two plane pancakes (as in 2D, not plain pancakes), of arbitrary shape can be simultaneously bisected by a single straight-line cute of a knife. See the figure

Although, I don’t have time now. I’ll prove this in an update as well has give you the three dimensional version, known as the Ham Sandwich Theorem. . .

Now aren’t you hungry?

UPDATE: Now here’s a proof:

Lemma 1: Given a direction [tex]0^{circ} leq theta leq 180^{circ} [/tex] in the plane, there exists a line with that direction that bisects the figure.
Proof: This follows from the fact that if you create a function that measures the area to one side of the line, it is continuous with respect to the location. Thus, there must exist a line that bisects the area.

Now consider any circle that contains both pancakes and consider the diameter with direction [tex]theta[/tex] for [tex]0^{circ} leq theta leq 180^{circ}[/tex]. By Lemma 1, we can make cuts in each pancake that are perpendicular to this diameter. Define [tex]p(theta)[/tex] and [tex]q(theta)[/tex] as the distance from the center to where these cuts cross the diameter. Then let [tex]r(theta) = p(theta)-q(theta)[/tex].

Because, [tex]theta =0^{circ}[/tex] and [tex]theta=180^{circ}[/tex] are opposite directions we have that [tex]p(0^{circ}) = – p(180^{circ})[/tex] as well as [tex]q(0^{circ}) = – q(180^{circ})[/tex] . Thus, [tex]r(0^{circ}) = -r(180^{circ})[/tex].

Now if [tex]r(0^{circ})=0[/tex], we are done and have bisected both pancakes. If [tex]r(0^{circ}) neq 0[/tex] then [tex]r(theta)[/tex] changes signs over [tex]0^{circ} leq theta leq 180^{circ}[/tex] and since [tex]r[/tex] is continuous, there exists some [tex]theta[/tex] such that [tex]r(theta)=0[/tex]. Q.E.D.

As I mentioned there is an extension to this theorem for higher dimension. It is generally refered to as the Ham Sandwich theorem, obtaining its name for the fact that given three solids (e.g., two slices of bread and a slice of ham), there exists a cut that bisects all three simultaneously (giving each two people and exact equal share of all three).


8 thoughts on “Favorite Theorem #1

  1. Thanks Alex (It is now repaired). I am still not very good at translating my thoughts to blog form. In fact, I had to type that proof three times since I accidentally deleted it twice while entering it into wordpress.

    Mom, come on now. I know you’ve wondered whether you could cut two pancakes in two equal portions with one straight cut. Now you know you can. Take that to the bank!


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