The Euler characteristic is a property of surfaces that remains the same under a topological homeomorphism.
It can be found by drawing a graph over the surface of the topological surface, and then counting the faces, vertices and edges. Then,
Originally, it was calculated for the polyhedra, which all have an Euler characteric of 2 because they exist as tilings of the sphere, which also has an Euler characteric of 2.
Changes Under Operations[]
The Euler characteristic of the disconnected sum of two surfaces is the sum of their Euler characteristics.
The Euler characteristic of the connected sum of two surfaces in the sum of their Euler characteristics, minus the Euler characteristic of the hypersphere (2, in the case of two-dimensional surfaces).
The Euler characteristic of the cartesian product of two surfaces is the product of their Euler characteristics.
Table of Euler Characteristics[]
| Surface | χ |
|---|---|
| Disk | 1 |
| Sphere | 2 |
| Torus | 0 |
| Double torus | -2 |
| Triple torus | -4 |
| Real projective plane | 1 |
| Möbius strip | 0 |
| Klein bottle | 0 |
| Dyck's surface | -1 |
| Sphere pair | 4 |
| Sphere triplet | 6 |