Corners, edges and faces

For a cube, $8 - 12 + 6 = 2$. For a tetrahedron, $4 - 6 + 4 = 2$. For a square pyramid, $5 - 8 + 5 = 2$. Leonhard Euler noticed in the 1750s that

$$V - E + F = 2$$

for every convex solid with flat faces, however many corners it has.

Here is the idea of a proof. Take away one face and stretch what is left flat on a table, like a net. Now remove edges one at a time. Removing an edge that separates two faces lowers $E$ and $F$ by one each; removing an edge that dangles at a corner lowers $E$ and $V$ by one each. Neither move changes $V - E + F$. At the end only a single point is left, with $1 - 0 + 0 = 1$; adding back the face we took away gives $2$. Making each step watertight takes real care — that care is the story of Imre Lakatos's book Proofs and Refutations.

The picture-frame solid has a hole through it, and for it $V - E + F = 0$. A solid with two holes gives $-2$. The number does not depend on how the surface is cut into faces — only on its shape up to stretching and bending, without tearing or gluing. Mathematicians call it the Euler characteristic $\chi$; for a closed surface with $g$ holes, $\chi = 2 - 2g$. It is one of the first topological invariants, and it reaches surprisingly far: the Gauss–Bonnet theorem says that the total curvature of a closed surface is $2\pi\chi$.