Vertices minus edges plus faces

A block built from small cubes, with a grid drawn on its surface: the gold points are the vertices V, the ivory lines the edges E, the cells the faces F, and the corner counts them one by one. Raise k with ‘Split each square k × k’ and the grid grows finer: V, E and F all climb, and V − E + F does not move; choose ‘Cut into triangles’ and each square gains one edge and one face, and still the number does not move. Now raise g with ‘Holes g’ and square tunnels are punched through the block: with no hole it is 2, like a sphere; with one hole, like a doughnut, it is 0; every further hole takes away 2 more: V − E + F = 2 − 2g, the χ in the corner. Polyhedra pierced like this are the ones Lhuilier counted in 1813.