This past Thanksgiving break, I had the honor of joining Josh G. at his house in NYC, where we feasted for breakfast on some incredibly delicious pumpkin waffles made by Josh’s mom. Unfortunately we soon came to the very last waffle (they were quite delicious)— with me, Josh, and Josh’s sister all ardently laying claim to a piece of it. Of course, we neatly sliced it into nearly-thirds — but they clearly weren’t *even* thirds.

Thus began the race for Proof of Equal Trisection of Waffles, where Josh and I tried to figure out how to properly trisect the waffle, using only classical implements of geometric construction (additionally, available in Josh’s kitchen), so that everyone got an equal slice. Turns out this is pretty hard. In fact, we later discovered that his has been proved to be impossible!

“Angle trisection is the division of an *arbitrary* angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836).”^{[1]}

Well, in this case, the proof (or would-be proof) is in the waffle-maker:

[1] via Wolfram’s Mathworld: http://mathworld.wolfram.com/AngleTrisection.html

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Wait, what shape was the original waffle? The impossibility proof is for trisecting an arbitrary angle with straightedge and compass, but there are many other things that can be trisected with those tools. For example, a line segment can be trisected:

http://www41.homepage.villanova.edu/robert.styer/trisecting%20segment/

If your waffles are originally rectangular, you can trisect them by constructing a third of one side and then cutting them into thirds.

If your waffles are originally circular, you can trisect them if you can construct a 120° angle (if you aren’t given the center point of the waffle, you can also construct it). One argument that you can construct 120° is that you can construct an equilateral triangle and then bisect one angle to get 30°. You can also construct a 90° angle. Constructing each of these in opposite directions relative to a line bisecting the waffle gives line segments along which you can cut to get a 120° slice of the waffle, or one third of a circular waffle. There is probably a much easier way, but I haven’t done any constructions in a long time.

To be honest, we didn’t even think of rectangular slices, proably because the original (unfair) distribution more closely resembled the pie-style slicing above.

We did try to invent the method for trisecting a line segment.

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I think this might be a confusion about the exact statement of the impossibility of the problem — it’s impossible in the general case, not in every case. If T(x,y) means that x is a method to trisect object y in classical Euclidean geometry using compass and straightedge, then

∃m:¬∃o:T(m,o)

is true (for example let o=the angle with measure 60°; no method m can trisect this angle) but

∀o:¬∃m:T(m,o)

is false (for example let o=the angle with measure 90°; any method to construct a 30° angle, such as bisecting a corner angle of an equilateral triangle, can construct this angle).

Oh, awesome — thanks for the clarification, Seth!

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Er, why not make rectangular slices instead of triangular ones? Problem solved.

Yeah, I guess those pumpkin waffles had us thinking unclearly.

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