The orbifold shop

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 ?