Math Monday: Bagel Cutting Revisited
DECEMBER 6, 2010

by George Hart

It has now been one year since I started writing these Math Monday columns on MAKE for The Museum of Mathematics. Thinking back to the first column on cutting linked bagel halves, I thought an appropriate anniversary column would show yet another interesting way to cut a bagel. Slicing on a slanted plane which is tangent to the surface at two places reveals a geometric surprise.

Bagel Circles 1

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.

Bagel Circles 2

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.

Bagel Circles 3

The proper position and slant of the slice will depend on the size of the bagel’s hole. As the above side-view shows, the slicing plane (red) must be chosen so it is tangent to the bagel at two places.

This article first appeared on Make: Online, December 6, 2010.

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