Suppose we have a symmetric pattern in the plane. We can make a symmetric map by subdividing the quotient orbifold into polygons, and then `unrolling it' or `unfolding it' to get a map in the plane.
If we look at a large area 
in the plane, made up from 
copies of a
fundamental domain, then each face in the map on the quotient orbifold
contributes faces to the region. An edge which is not on a mirror
also contributes approximately copies - approximately, because when
it is on the boundary of , we don't quite know how to match
it with a fundametnal region.
In general, if an edge or point has order 
symmetry which
which preserves it, it contributes approximately 
copies of
itself to , since each time it occurs, as long as it is not on
the boundary of , it is counted in copies of the fundamental domain.
Thus,
copies.
vertices to .
it contributes approximately 
vertices.
.
corner reflector, it contributes
approximately 
Question.
Can you justify the use of `approximately' in the list above?
Take the area 
to be the union of
all vertices, edges, and faces that intersect a disk of radius 
in the plane, along with all edges of any face that intersects and
all vertices of any edge that intersects. Can you show that the
ratio of the true number to the estimated number is arbitrarily close
to 1, for high enough?
Definition.
The orbifold Euler characteristic is 
, where each
vertex and edge is given weight 
, where is the order of symmetry
which preserves it.
It is important to keep in mind the distinction between the topological
Euler characteristic and the orbifold Euler characteristic. For instance,
consider the billiard table orbifold, which is just a rectangle.
In the orbifold Euler characteristic,
the four corners each count 
, the four edges count 
, and the
face counts 1, for a total of 0. In contrast, the
topological Euler characteristic
is 
.
Theorem. The quotient orbifold of for any symmetry pattern in the Euclidean plane which has a bounded fundamental region has orbifold Euler number 0.
Sketch of proof: take a large area in the plane that is topologically
a disk. Its Euler characteristic is 1. This is approximately
equal to times the orbifold Euler characteristic, for some large ,
so the orbifold Euler characteristic must be 0.
How do the people at The Orbifold Shop figure its prices?
The cost is based on the orbifold Euler characteristic: it costs
to lower the orbifold Euler characteristic by 1.
When they install a fancy new part, they calculate the difference between
the new part and the part that was traded in.
For instance, to install
a cone point, they remove an ordinary point. An ordinary point counts 1,
while an order cone point counts , so the difference is 
.
To install a handle, they arrange a map on the original orbifold so that it
has a square face. They remove the square, and identify opposite edges
of it. This identifies all four vertices to a single vertex. The net
effect is to remove 1 face, remove 2 edges (since 4 are reduced to 2),
and to remove 3 vertices. The effect on the orbifold Euler characteristic
is to subtract 
, so the cost is 
.
Question. Check the validity of the costs charged by The Orbifold Shop for the other parts of an orbifold.
To complete the connection between orbifold Euler characteristic and symmetry patterns, we would have to verify that each of the possible configurations of parts with orbifold Euler characteristic 0 actually does come from a symmetry pattern in the plane. This can be done in a straightforward way by explicit constructions. It is illuminating to see a few representative examples, but it is not very illuminating to see the entire exercise unless you go through it yourself.