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:
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.
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.
To sum it all up, once we know what all Euler Characteristic 0 orbifolds are, we know what all planar patterns with finite motif can be.
It's easy-- but tedious-- to check there are only 17 ways of assembling such an orbifold at the orbifold store, and so can only be 17 planar patterns with finite motif.
But we will leave two important parts of the overall proofunfilled: First, we didn't really prove that the Euler characteristic of a sphere must be 2. All the other Euler characteristics rested on that statement. Second, we didn't really justify that no more kinds of orbifold part are available. But up to those statements, we're done! And for those of you who have seen the classification of planar patterns in abstract algebra, you probably agree this is a much more sensible approach! ---- cgs
A Field Guide to the Orbifolds excerpted from the Geometry and Imagination Course,
complete with scrawled notes in the hand of John Conway.
Chaim Goodman-Strauss Dept. Mathematics Univ. Arkansas Fayetteville, AR 72701 email@example.com 501-575-6332