Euler Characteristic

Königsberg City Map and 7 Bridges
(Königsberg is a city of Prussia where Goldbach (Yes, the Goldbach Conjecture!) is born.)





The Swiss mathematician Leonhard Euler is one of the great mathematicians of all times. He used to live in a city called, Königsberg, which is a city divided into a few islands by several rivers. See the picture. It was at there that he put forward the Königsberg bridges problem (i.e. the sales man problem): One cannot cross all seven bridges just once for each.
TWF also made up a story for Euler to find the Euler characteristic formula, V - E - F. It goes like this... There used to be an isolated island in the city and there was no brige connecting it with the rest of the city. But one day, the city finally decided to build a brige to the island and Euler saw the construction. Then he thought, the number of the isolated islands is decreased by one due to the new bridge. Everybody can see this... Well, Euler did one step further: The bridge was acting as a negtive bridge. So, # islands - # bridges. One can another step further.. If there is a dock builded between the two bridges, the dock is then acting as a negative bridge, i.e. a positive island, # islands - # bridges + # docks. So comes the Euler characteristic V - E - F, where V stands for the number of vertices of a polyhedra, E stands for the edge (connecting vertices) and F for the face (connecting edges). Well, the story does not stop here if you imagine a higher dimensional world than 3-D. You may add some structure to connect two faces in a 4-D world and the formula repeats the alternative sigh pattern. The Euler characteristic is applied into counting the size of categories...
Just like this idea. Simple and clear.