Diagram 6: Fundamental polygon
A fundamental polygon is one with edges labeled by letters and arrows, such that each letter occurs twice. The idea is to think of edges labeled with the same letter as "essentially the same". One could just glue those together, but in 3d this won't always work out nicely without self-intersections.
The Picture below shows the fundamental polygon for the Klein bottle which cannot be embedded in our 3d. Our diagrams have been invented to fully replace the need for the power to imagine complicated surfaces. They are halfway to a purely algebraic representation of using just a word of letters and antiletters (inverse letters).
Lets play with those and, say, start with any polygon and attach labels and arrows like random. What we get is called a combinatorial surface. Now there is to wonder wether the corresponding topological surface really exists. Or what happens when we introduce more polygons into the game.
Answers can be found in Norman Wildbergers AlgTop18. The lecture shows a proof that all topological surfaces can be classified (this is how fundamental theorems start!) by counting the number of handles or crosscaps (you never need both for any given surface).
For this he does diagrammatic computations on the blackboard using two operations: Collapsing a cone, and, cut & paste. I'm inviting you to invest that hour of your time to experience the beauty of diagrams in action and learn how mathematicians handle unimaginaries.
- One can look for fundamental polygons in any tiling (not necessarily of the plane).
- Another fun thing to do is to check that the operations preserve the Euler charakteristic (also in the lecture).
- You might want to stare at the fundamental region of the Klein quartic, now that you know how to read it.