The Fundamental Polygon

Last week we received a couple questions on homework problem 1.3 regarding whether a constructive proof is necessary, i.e., did you need to actually come up with examples of triangulations that achieve the desired bounds? To reiterate what was said in the comments, the answer is no: you simply needed to show (algebraically) that these bounds hold. But if you were looking for a constructive proof, a nice tool to know about is something called the fundamental polygon. The basic idea is that a surface of any genus can be cut into a single disk, which can be visualized as a polygon in the plane. Visualizing a triangulation (or other data) on the fundamental polygon is often much simpler than visualizing it on the embedded torus — consider this example, for instance:



Here the arrows mark identifications of edges — for instance, the top and bottom edges get “glued” together along the positive \(x\)-axis. (Try convincing yourself that these gluings really do produce a torus — note that at some point you’ll need to make a \(180^\circ\) “twist.”) One way to visualize the identifications is to imagine that the fundamental polygon tiles the plane:

The torus is interesting because it actually admits two different fundamental polygons: the hexagon and the square (corresponding to two different tilings of the Euclidean plane). So we could also visualize the torus on the square, leading to an even simpler triangulation:



(By the way, is this really what you’d call a triangulation? Each region certainly has three sides, but each “triangle” has only one vertex! What about the first example? There all triangles have three vertices, but they share the same three vertices. So combinatorially these triangles are not distinct — in other words, this tessellation cannot be described via a simplicial complex. What’s the smallest simplicial decomposition you can come up with for the torus?)


In general the fundamental polygon for a torus of genus \(g\) has \(4g\) sides with identifications \(a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_n b_n a^{-1}_n b_n\), where two edges with the same letter get identified and inverse just means that the edge direction is reversed. For instance, the fundamental polygon for the double torus looks like this:



(Note that all the polygon vertices ultimately get identified with a single point on the surface.) From here it becomes easy to start playing around with tessellations — for instance, here’s how you decompose a surface of any genus into quadrilaterals, using only two irregular vertices (can you do better?):



Tiling the Euclidean plane with the fundamental polygon is impossible for a surface of genus \(g \geq 2\), since the interior angles of the fundamental polygon don’t evenly divide \(2\pi\) (proof!). Fortunately, we can still tile the hyperbolic plane, i.e., the plane with constant negative curvature. For instance, here’s a tiling of the hyperbolic plane by octagons:



From here there are all sorts of fascinating things to say about covering spaces, uniformization, and especially the fundamental group of a surface — if you’re interested I highly recommend you take a look at Allen Hatcher’s (free!) book on algebraic topology.

2 Responses to “The Fundamental Polygon”

  1. ajerves says:

    Keenan, please, whenever you have some free time, could we please talk about it, seems very interesting for me, but I don’t think I’m fully understanding what you guys are trying to say! Thank you!