A Mathematical Approach to Slicing Doughnuts

Figure 1 shows a torus, a surface studied by mathematicians not just because it reminds them of a chocolate doughnut or a bagel filled with smoked salmon and cream cheese, but also because it has useful applications and interesting mathematical properties. We are about to illustrate just one such property (Well - we think it's interesting). The axes shown in red in the diagram just indicate how we have placed the torus in the usual coordinate system for three dimensional space.

Authors John Banks Jeff Brooks Further Reading School of Mathematics La Trobe University.

Figure 1: A Garden Variety Torus

What we propose to do is slice the torus through with a plane - much as you might cut a doughnut in half with a large knife - and have a look at the "faces" of the cut. However, instead of just cutting vertically, we will try cutting at various different angles. The only restriction is that the plane we slice with must contain one of the horizontal axes shown in the diagram. For the sake of convenience we will use the axis pointing toward the viewer in the diagram.

Figure 2: The Simplest Slice

Figure 2 illustrates what happens if you slice the torus in the way you would probably slice your doughnut if you wanted to share it with a friend. Assuming you wanted to share it fairly, you would just slice it vertically throug the middle. It is easy to see (and even check with a real doughnut) that the faces of the cut are just a pair of circular disks. Now, being mathematicians, we like to throw away all of the unnecessary features and just deal with essentials. So we assume that the torus has no interior. It's just a hollow shell with no thickness. A doughnut like that would leave you hungry, but mathematically speaking, that's what a "torus" actually is. A torus with its interior included is properly called a "solid torus". Anyway, the "faces" of our vertical cut are just circles now and we use the technical term "intersection" to describe them.

Figure 2 also shows what how we intend to vary the angle of the cut by rotating the plane about the axis pointing toward the viewer. Figure 3 shows the cut where the plane is horizontal. This one may be a little difficult in practice if your doughnut is soft, but it is easy to see that the intersection consists of two concentric circles.

Figure 3: Another Easy Slice

So now let's watch what happens to the shape of the cut as we rotate the plane through 360 degrees. This is what is happenning in figure 4 (This won't make much sense if your web browser doesn't support animations). Starting from the horizontal position where there are two concentric circles we go through the vertical position where there are two separate circles and then back again. We are viewing the plane of intersection at right angles at every stage ogf the animation. Observe that the two separate circles gradually deform into kidney shapes and then, somewhere along the line, they cross over into the concentric shapes which eventually become the concentric circles at the horizontal position.

Figure 4: Varying the Angle of the Cut