Saturday, September 17, 2022

Pizza theorem

In Acheson's The Wonder Book of Geometry, I came across the "pizza theorem."

The basic form of the problem says to take any point within a disk, and construct a grid of chords which all cross at that point. There are two pairs of perpendicular chords, evenly spaced at 45 degrees.

Alternate radial sections are shaded. The result is that the pizza is always evenly divided between light and dark pieces, even though the pieces themselves are oddly shaped.

Here is Acheson's figure.

If the grid coincides with the center of the circle, the theorem is obviously correct. And if the grid moves by sliding along a diagonal of the circle, for example vertically, the result is still correct. That's because the figure has mirror-image symmetry.

After a vertical move to move to the desired final radial position, the grid can be either rotated or moved "horizontally", parallel to a chord but not on a diagonal of the circle. I worked out a geometric proof that this movement does not change the relative areas.

The basic idea is that the areas gained and lost by the movement go like the difference t-s for the two parts of each chord. (Except that the area covered by the vertical chord is fatter by sqrt(2)).

But that difference is closely related to the distance from the center of the chord to the grid center. The sum of those is easily shown to be invariant, because the center of each chord lies on its vertical bisector, which passes through the center of the circle.

I also found on the web a nice geometric proof for invariance under rotation (first answer at the link). I expanded it to make it (hopefully) even clearer (pdf).

I also added a chapter to my geometry book (38 MB). There are shorter versions elsewhere that I linked previously, but they are now outdated.