In certain areas of mathematics, a sphere attached to a sphere is still a sphere, though perhaps a bigger or lumpier one. And if a sphere gets glued onto a doughnut, you still have a doughnut — with a blister. But if two doughnuts merge together, they form a two-holed shape. To mathematicians, that’s something else completely. That quality makes spheres a crucial test case for geometers. Mathematicians can often transfer lessons learned on spheres to more complex shapes by looking at what happens when you sew the two together. Included in this all-around look at spheres is a comment from John Etnyre, professor in the School of Mathematics.