**Excerpted from Geometry
and the Imagination by John Conway, Bill Thurston, Jane Gilman and Peter Doyle. These notes are
really really great! Annotations in italics and Math 5337 links at
the end:**

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: .
*a.k.a. a TUBE* - Mirror: .
- Cross-cap: .
*We were sloppy about this one: denoted***o**; think of sewing in a mobius band along it's single edge. - Order cone point: .
- Order corner reflector: .
*a.k.a. fold corner*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.

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

*Q9* Check that some of the orbifolds you've found can be bought
at the Orbifold Shop

Peter Doyle

Euler Characteristic of an Orbifold

to outline

Chaim Goodman-StraussDept. Mathematics Univ. Arkansas Fayetteville, AR 72701strauss@comp.uark.edu501-575-6332