7 Bridges

Jack Dikian

January 2011

While writing about Knot Theory last night the subject and associated problems in Topology bought back many memories that are reminiscent to a child experiencing the color and movement of a circus for the first time.

Topology and in particular Graph Theory is often not part of high school math curricula, and thus for many it sounds strange and intimidating. However, there are some readily graspable ideas at the base of topology that are interesting, fun, and highly applicable to all sorts of situations. One of these is the topology of networks, first developed by Euler in the early 1700’s inspired by the Seven Bridges of Konigsberg.

Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in Russia. The city is built around the River Pregel where it joins another river. An island named Kniephof is in the middle of where the two rivers join. There are seven bridges that join the different parts of the city on both sides of the rivers and the island.

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. No one was able to find a way to do it, and this came to the attention of a mathematician Euler.

In 1735, Euler, before the Russian Academy presented the solution to the problem explaining why crossing all seven bridges without crossing a bridge twice was impossible.

Euler simplified the bridge problem by representing each land mass as a point and each bridge as a line. He reasoned that anyone standing on land would have to have a way to get on and off. Thus each land mass would need an even number of bridges. But in Konigsberg, each land mass had an odd number of bridges. This was why all seven bridges could not be crossed without crossing one more than twice.

Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.