Euler characteristic

The Euler characteristic is a special number that describes how the parts of a shape (like a polyhedron) fit together. For many solid shapes, if you count the number of vertices ($V$), edges ($E$), and faces ($F$), the Euler characteristic is given by the formula: $V - E + F$ For most simple polyhedra (like cubes and pyramids), this value is 2.

Important properties

• For convex polyhedra, $V - E + F = 2$.

• The Euler characteristic helps check if a polyhedron is 'well-formed' (e.g., not twisted or with holes).

• If you change a polyhedron by adding or removing parts without making holes or connecting pieces, the Euler characteristic stays the same.