Chapter 4.7, Problem 17MS

Doughnuts in dimensions. Suppose we have a mysterious object in four dimensions. If we slice the object in half with a 3-dimensional slice, we’d see the surface of a hollow doughnut.

If we take a 3-dimensional slice just above or below that first slice, we’d see the surface of another doughnut-this one thinner than the first.

As we slice higher (and lower) we see doughnuts whose waistlines get thinner and thinner, until finally we just see a circle.

What is the 4-dimensional object if we know the level 3-dimensional slices? Why is the answer “a circle of spheres”?