1. The Seven Bridges of Konigsberg QuickTime™ and a H.264 decompressor are needed to see this picture.
2. It is now called KaliningradCan you see the seven bridges?
3. The people wondered whether or not one could walk around the city in a way thatwould involve crossing each bridge exactly once.Problem 1Try it. Sketch the above map of the city on a sheet of paper and try to plan yourjourney with a pencil in such a way that you trace over each bridge once and only onceand you complete the plan with one continuous pencil stroke
4. Can’t do it - neither could Euler - a very famous mathematician. In fact he provedthat it couldn’t be done.
5. Leonhard Euler QuickTime™ and a decompressor are needed to see this picture.
6. Problem 2Suppose they had decided to build one fewer bridge in Konigsberg, sothat the map looked like this: Now try and solve the problem
7. Problem 3Does it matter which bridge you take away? What if youadd bridges? Come up with some maps on your own, andtry to plan your journey for each one Can you draw any conclusions?
8. These are the same Diagram
9. Node (Vertice) Edge (Arc)A network is a figure made up of nodes and edges A node is ODD if it has an odd number of edges leading into it otherwise it is called even An Euler path is a continuous path that passesthrough each arc once and only once - we say the network is transversable
10. Euler proved:If a network has more than two odd vertices,it does not have an Euler path i.e it is nottransversableHe also proved:If a network has two or zero odd vertices, ithas at least one Euler path. In particular, if anetwork has exactly two odd vertices, thenits Euler path can only start on one of theodd vertices Why is this important..... Circuits?
11. This branch of Mathematics is called Graph Theory or more specifically topological graph theoryVery useful for proving the “Hairy Ball Theorem” QuickTime™ and a decompressor are needed to see this picture.
12. Just to alter your perception of reality a bit... QuickTime™ and a decompressor are needed to see this picture.