8 − 12 + 6 = 2. Count vertices minus edges plus faces on a cube and you get 2; do it on a tetrahedron and you get 2 again, and every convex polyhedron returns that same number. The Euler characteristic χ survives any continuous deformation that neither tears nor glues, so it sees what a ruler cannot: a sphere has χ = 2, a torus χ = 0, a two-holed surface χ = −2, each extra hole costing two. That is why a handled mug and a doughnut are one object to a topologist, while a Möbius band has just one side and a single boundary curve.
8 − 12 + 6 = 2. Count vertices minus edges plus faces on a cube and you get 2; do it on a tetrahedron and you get 2 again, and every convex polyhedron returns that same number. The Euler characteristic χ survives any continuous deformation that neither tears nor glues, so it sees what a ruler cannot: a sphere has χ = 2, a torus χ = 0, a two-holed surface χ = −2, each extra hole costing two. That is why a handled mug and a doughnut are one object to a topologist, while a Möbius band has just one side and a single boundary curve.
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