The Orbifold Shop has gone into the business of installing orbifold parts. They offer a special promotional deal: a free coupon for $2.00 worth of parts, installation included, to anyone acquiring a new orbifold.

There are only a few kinds of features for two-dimensional orbifolds, but they can be used in interesting combinations.

- Handle: .
- Mirror: .
- Cross-cap: .
- Order cone point: .
- Order corner reflector: . Prerequisite: at least one mirror. Must specify in mirror and position in mirror to be installed.

With the coupon, for example, you could order an orbifold with four order 2 cone points, costing each. Or, you could order an order 3 cone point costing , a mirror costing , and an order 3 corner reflector costing .

**Theorem.**
If you exactly spend your coupon at the Orbifold Shop, you will have
a quotient orbifold coming from a symmetrically repeating
pattern in the Euclidean plane with a bounded fundamental domain.
There are exactly different ways to do this, and corresponding
to the different symmetrically repeating patterns with bounded
fundamental domain in the Euclidean plane.

**Figure 20:** This is the pattern obtained when you
buy four order 2 cone points for each.

**Figure 21:** This is the pattern obtained by
buying an order 3 cone point, a mirror, and an order 3 corner reflector.

**Question.**
What combinations of parts can you find that cost exactly ?