math


By George Hart for the Museum of Mathematics

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.

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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.

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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.

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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.