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

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

Original links:

Next: The Euler characteristic Up: Geometry and the Imagination Previous: Stereographic Projection

Peter Doyle

Links for Symmetry & the Shape of Space

```
Chaim Goodman-Strauss
Dept. Mathematics
Univ. Arkansas
Fayetteville, AR 72701
strauss@comp.uark.edu
501-575-6332```