Outline of the proof that there are exactly 17 symmetry types for
repeating patterns in the plane.
- 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
- There are exactly 17 orbifolds whose features cost exactly $2,
and so exactly 17 symmetry types for repeating patterns in the plane.