The Seven Bridges of Konigsberg was an option we could investigate in either one of the dailies or one of the hand outs on Euler. I think the main reason why I found this problem interesting was that it shows how mathematical theory can apply to the "real world" and further shows the brilliance of Euler.

So, Konigsberg was a major city for commerce and trade in the 13th century, in what is now modern day Russia, that was divided into four distinct districts by the Pregel river. To connect all the land seven bridges were made. Apparently, it is said that the people living there at the time always used to play a game where they would try to see if they could travel around the city to each of the four districts while only going over each of the seven bridges once. For reference, here is a picture of what the city and bridges looked like:

In 1736, Euler proved that there was no solution to this problem. In other words, there is no way for someone to walk around the entire city and only go over each bridge once. The real beauty of this problem is the way in which Euler solved it, or at least to me. In class we discussed how he always had unique ways of thinking about and solving problems and this one is no different. He realized that only the sequence of bridges mattered and not the actual route inside each land mass. Because of this, he was then able to construct a graph of the problem, with the land masses as endpoints and the bridges as the connecting lines as seen below:

Now with this new representation of the problem, it is fairly easy to see that each land mass has an odd number of bridges connected to it, either 3 or 5. Euler then concluded that there is no way to only use each bridge once no matter where you start or end.

Overall, this isn't too complicated of a problem, especially in today's world of mathematics, but at the time the significance of it was huge. For one, it is now considered the first theorem in graph theory and the first true proof in the theory of networks. Also, the way he thought of the problem was so innovative at the time. Unlike most, he realized that the key information in the problem was not the exact position of the bridges and land but rather how many there were and their endpoints.

Personally, I just find it amazing how one man could have such an insight into so many different aspects of mathematics. I always knew about the impacts he had in other fields of mathematics like Euler's formula or Euler's identity, but I didn't know that he basically came up with graph theory as well. I know we talked about how great he was in class but I didn't really appreciate or believe it until looking in to this and a few other problems he solved.

I like how you used the Konigsberg city as an example in the beginning. It gives readers background info as well as, like you said, illustrates how math can be used to solve problems in the world. The only thing I could think of that you could add is how would the bridges need to be oriented (add more or less brides) to make the problem possible.

ReplyDelete*less bridges

DeleteGood examination of a single problem. Why does the odd degree of the vertices mean there is no path? (content) Other Cs +

ReplyDeleteAmazing to think of how Euler just thought mathematically.