Orbifold Euler Characteristic for Planar Tilings
Suppose T is a tile for a wallpaper pattern in the plane. Mark any mirrored sides and label
all the centers of rotational symmetry. Think of it as built from vertices, edges, and faces,
where each center is a vertex.
Our goal is to use these markings on a single tile to figure out how the vertices, edges, and
faces add up when the plane fills up with copies of the tile. The idea (as we discussed in
class) is that when a symmetry is applied to the tile to move it somewhere else, some pieces
of the ‘new’ tile overlap similar (or identical) pieces of the old tile. For instance, one edge
of the original tile might line up with some other edge of the adjacent tile, so that all we
really see is a single edge in the tiled plane, even though it’s made out of two edges of the
tile itself. Thus we should count each such edge only as 1/2 an edge.
So we should consider a single tile T and ask ourselves how many different symmetries move
that tile in such a way so as to overlap itself somewhere, and then we need to make sure we
reduce our count appropriately.
FACE: For the single face of the tile, there aren’t any symmetries that make it overlap with
itself, so it counts as a full face.
NON-MIRRORED EDGE: If e is a non-mirrored edge, then there must be some other edge
e! on the tile that moves via a symmetry so as to overlap it. So we could count each of e and
e! as a half edge, so that together they make a single edge. But there’s another way to think
about this. We could instead pretend our tile is made of rubber and go ahead and bend it
around, gluing e to e! . If we go ahead and do this for all non-mirrored edges, then we don’t
have to do any funny counting for these, because any pairs that will get matched up have
already been. (This is what I forgot to do in class; I was counting non-mirrored edges as
entire edges, but without gluing them up first.)
MIRRORED EDGE: These edges don’t move at all when we apply the symmetry that reflects
across them, so we can’t do the trick we did for non-mirrored edges (gluing the edge to itself
without moving anything doesn’t really do anything). So we really do have to count mirrored
edges as half-edges.
REGULAR VERTEX: By a regular vertex I mean a vertex that is not a center of rotation.
If we glue up the non-mirrored edges as described above, these will simultaneously be glued
up appropriately also, so that we can then count them normally.
CONE VERTEX: By this I mean a vertex that is the center of a rotational symmetry, and
has non-mirrored edges on both sides of it. There is an angle of the form 2π/n at such a
vertex. The crystallographic restriction theorem implies that n is either 2, 3, 4, or 6. The
rotation around such a vertex move the edge on one side of the vertex to the other. If we
imagine our tile made of paper, we can glue these edges together and it will make a cone
shape with our vertex at the top. That’s why it’s called a cone vertex. If the angle is 2π/n,
then there are n different symmetries that don’t move this vertex at all, so we should count
it as 1/n of a vertex.
DIHEDRAL VERTEX: By this I mean a vertex that is the center of a rotational symmetry
and has mirrored edges on both sides. Such a vertex is not moved by reflection about either
one of the sides, and we can repeat this. (Imagine taking a triangular slice of quesadilla and
flipping it over and over across opposite angular sides.) Two of these flips will be a rotation
by 2π/n as before, but the flips double how many symmetries don’t move it. Thus we should
count such a vertex as π/n of a vertex.
ORBIFOLD EULER CHARACTERISTIC: The formal definition is that we count faces,
edges, and vertices as described above (after having done the appropriate gluing), and then
add them up with alternate signs.
THE OTHER FORMULA: Look at the glued up tile and ignore the labelling: just count
up the faces, edges, and vertices, not using any fractions (obviously some fractions will be
taken care of automatically by the gluing – that’s really the whole point here). The question
now is how is this unlabeled Euler characteristic different from what we get when we put
the labels on? The first thing to note is that the largest the Euler characteristic can be is
two (for a sphere), so let’s start there.
Now we just go through all the alternatives and see what happens. The idea is that we
discover that the very first formula we saw in class is what works. For example, if a vertex
is actually a cone vertex labelled n, then instead of counting as 1, it counts as 1/n. Thus
changing it from unlabelled to labelled subtracts (n − 1)/n. If there are any mirrored edges
at all, then instead of a sphere (which has Euler characteristic 2), we will have a disk, which
has Euler characteristic 1. Thus the presence of any mirrored edge requires we subtract 1.
The other cases are similar, leading to the formula that starts with two an subtracts one if
it’s all translations, one if there’s a mirror, one if there’s a glide reflection, (n − 1)/n for each
cone vertex labeled n, and (n − 1)/2n for each dihedral vertex labeled n.
EXAMPLES: The other pages show some examples of this calculation. Each example has
three rows. In each case, the first row consists of: the marked tile; the tile with non-mirrored
edges appropriately identified (i.e., the orbifold); the ordinary (unlabeled) Euler characteristic of the orbifold; a small piece of the tiling pattern. The second row is a calculation
of the orbifold characteristic directly, as described in the steps above. The third row is a
calculation of the orbifold characteristic using ‘the other formula.’ You should look through
the examples and make sure you know where all the numbers in all the calculations come
from. Note that on the orbifolds the cone vertices are blue, the dihedral ones are red/orange,
and the mirrors are green.