From 2009 to 2013, Make: ran a very popular column here on the site called “Math Monday.” Written by George Hart, of the then newly-formed Math Museum, the idea was to present a very fun, hands-on, and thought-provoking series of activities exploring various aspect of mathematics. The column (taken up several years in by Glen Whitney after Hart left the museum) explored such things as making geometric objects from playing cards, business cards, and office supplies, understanding tensegrity structures, math in fashion, fun with fractals, and other cool and accessible explorations of mathematics that might appeal to makers.
We are going to be reviving these columns and be reposting them on Make:. Look for future “Math Monday” reruns each Monday here on on Make:.
One of the most popular subjects that George covered in his column was literal bagel hacking, exploring different mathematical ideas through bagel slicing. Here are those four columns.
Start your day right by making this challenging bagel cut, and see if you’re really awake yet. Can you figure out how to slice a bagel into two congruent halves which pass through each others holes, like two links of a chain? Hint: The motion of the knife follows the surface of a two-twist Mobius strip. If you hack up a dozen bagels and still haven’t solved the puzzle, you can check out the instructions here.
The bread of a bagel forms a simple loop, which mathematicians call “the unknot.” But there are two easy ways to cut a bagel into a simple overhand knot, or “trefoil” knot. Above is a what mathematicians call “the (2,3)-torus knot, toasted with cream cheese.” More on these bagel knots here.
The planar cross section is two overlapping circles called Villarceau circles after the French mathematician, Yvon Villarceau, who wrote about them in the mid 1800s. I’ve indicated them here with colored markers, but you can see they are not perfectly round on a real bagel, because of its flat bottom and other irregularities. On an ideal torus, this slanted slice gives two perfect overlapping circles. The proper position and slant of the slice will depend on the size of the bagel’s hole. The slicing plane must be chosen so that it is tangent to the bagel in two places. More info and images on the original post.
How can you slice a bagel into thirteen pieces with just three simultaneous planar cuts? Some of the pieces will have to be rather small, but the cuts are doable. Start with a slanted cut as steep as you can make it without keeping a hole in either the top or bottom piece. Full instruction here.