### 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
characteristic zero.
- There are exactly 17 orbifolds whose features cost exactly $2,
and so exactly 17 symmetry types for repeating patterns in the plane.

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