How Many Sides Does a Tire Have?

Abstract

How many sides does a tire have: two, three, or only one? The question sounds like a joke, but it hides a real modeling choice. We idealize a car tire as a solid torus and compare two ways of marking its surface. In the first, a rim is part of the cellular boundary of two chosen regions. In the second, the same rim is merely a closed loop on one connected surface. Following that small difference leads naturally to cells, boundary operators, and a useful warning: before topology can count anything, we must decide exactly what is being counted.