Cauchy's projection–triangulation proof of V − E + F = 2 (1811)

Frequently asked questions

Does the same argument work for a polyhedron shaped like a torus (a ring with a hole)?

No — the flattening step in the first step needs the surface to be a topological sphere. For a surface with $g$ holes (genus $g$ ), the same style of argument instead gives the general Euler characteristic $V - E + F = 2 - 2g$ , which equals $2$ only when $g = 0$ .

Must a polyhedron be convex for Cauchy's proof to apply, or is something weaker enough?

Something weaker is enough: the proof only needs the surface to be a topological sphere (star-shaped and even many dented, non-convex solids qualify), because that is exactly the property Step 1 uses to flatten it without crossings. Genuine convexity is a convenient sufficient condition, historically the one Cauchy assumed, but not the essential one.

Source

Augustin-Louis Cauchy, 1811

Proofs from THE BOOK (6th ed.)

Step 6 of 6: Beyond the sphere: why non-convex and toroidal solids can fail

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In plain words

A coffee mug and a doughnut are famously the "same shape" to a topologist because both have exactly one hole running through them; a solid built like a picture frame, with a rectangular tunnel cut through its middle, is secretly a doughnut wearing a disguise. The whole flattening trick from Step 1 needed the surface to have no holes at all, so a framed shape breaks the proof at its very first move, not at some later technicality.

\chi = V - E + F = 2 - 2g, \qquad \text{e.g. } g=1 \text{ (a hole through the solid)} \implies V - E + F = 0

Detailed analysis

The argument built in Steps 1–5 rests entirely on one assumption used at the very first move: the polyhedron's surface, once punctured, can be flattened onto the plane without any edges crossing, which is only possible when the surface was a topological sphere to begin with. Convex polyhedra always have this property, but so do many non-convex ones (a dented cube, say) — convexity itself is never used anywhere in the argument, only sphere-like topology.

Terms in this step

Genus: A whole number $g$ counting how many holes/handles a surface has: $g=0$ for a sphere, $g=1$ for a torus (doughnut), and so on.
Euler characteristic: The quantity $\chi=V-E+F$ computed from any polyhedral (or triangulated) surface; Steps 1–5 show it depends only on the surface's topology, not on its particular shape.