What is the Euler characteristic and why is it always 2 for convex polyhedra?

Euler's formula $V - E + F = 2$ is a topological invariant — it depends on the **shape** of the surface, not on the specific arrangement of vertices, edges, and faces. **Why 2?** The number $2$ is the Euler characteristic $\chi$ of the sphere $S^2$. Any convex polyhedron is topologically equivalent (homeomorphic) to a sphere. You can imagine inflating the polyhedron until it becomes a round sphere — the network of edges becomes a graph on the sphere, and the relation $V - E + F = 2$ holds. **For Non-Convex Polyhedra:** The formula changes if the polyhedron has **holes** (genus $g$): $$V - E + F = 2 - 2g$$ | Surface | Euler Characteristic $\chi$ | |---|---| | Sphere ($S^2$) | $2$ | | Torus (doughnut, $g=1$) | $0$ | | Double torus ($g=2$) | $-2$ | | Projective plane ($\mathbb{RP}^2$) | $1$ | | Möbius strip (with boundary) | $0$ | **Example — Torus:** A cube with a square hole drilled through it (a \"tunnel\") has $\chi = 0$. You can verify: if the inner tunnel adds 4 new square faces but removes some original faces, the relation adjusts to $V - E + F = 0$. **Generalisation to higher dimensions:** The Euler characteristic generalises to $n$-dimensional CW complexes as: $$\chi = \sum_{k=0}^{n} (-1)^k \cdot (\ ext{number of $k$-cells})$$ This is one of the most fundamental invariants in algebraic topology, linking combinatorics, geometry, and topology.