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Mathematics

Planar Graphs and Euler's Formula for Polyhedra

Quick fact

Euler's formula famously states that for any convex polyhedron (or any connected planar graph), V - E + F = 2. This single equation has powerful consequences, such as proving that a planar graph can have at most 3V - 6 edges, which can be used to show that K5 and K3,3 are not planar.

Why this is interesting

You've seen maps with many countries, but can you always draw them without edges crossing? What does that have to do with a simple arithmetic relation?