17 Symmetry Types, MA562

Repeating patterns give rise to orbifolds when all their symmetric points are brought together or "identified". Symmetries of the patterns correspond roughly to features of the orbifold.
The only features an orbifold can have are handles (o), cross-handles, cone points (n), boundaries (*), corner points (n), and cross-caps (x). A cross-handle can be constructed by combining two cross-caps.
Each of these features has a certain "cost". We can build any possible orbifold by adding features to a sphere (Euler characteristic 2) and each feature added reduces the orbifold Euler chracteristic by the cost of the feature.
The orbifolds that correspond to symmetry types of repeating patterns on the plane are exactly those with orbifold Euler characteristic zero.
There are exactly 17 orbifolds whose features cost exactly $2, and so exactly 17 symmetry types for repeating patterns in the plane.