Königsberg Bridges

 

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There are several branches of topology which deal with different problems and ideas. The work which is considered to be the first work in topology was Euler's paper on Königsberg bridge sproblem. It was published in 1736 and was entitled Solutio problematis ad geometriam situs pertinentis : translated into English this is The solution of a problem relating to the geometry of position.

Königsberg is a town which has a river going through its middle and a number of bridges between the parts of the town. Euler illustrated this as follows

The river Pregel divides the town into four sections - A, B, C, and D. There are seven bridges, and some of the town's citizens apparently wondered whether they can walk across all seven bridges without crossing any bridge more than once.   Euler in his paper explained that such a journey was impossible by studying the network of bridges. He concluded that, if a journey would begin at one land mass and end at another, than these two could have an odd number of connecting bridges, while each other land mass should have an even number of bridges connected to it otherwise the journey is not possible.

Try to draw a diagram which represents Königsberg's bridges problem by representing land masses by points and bridges by lines that connect them. You are now using one kind of topological diagram which is called "planar graph".

Try to cross the bridges only once but to cover each bridge and to arrive at all land masses. The problem is to draw this picture without going over any line twice, and not lifting your pencil from the paper.

If you look carefully, you will see that all four vertices (points) in the picture above have an odd number of lines which connect them (these are by the way called arcs). Every vertex with an odd number of arcs attached to it has to be either at the end or the beginning of the pencil path. Try to prove or disprove this!

Because your 'journey' has to start and finish, the vertices with odd number of paths have to be either at the beginning or an end of your journey.

Euler generalised this thinking into a number of definitions followed by a theorem:

Definition 1: A network is a figure made up of vertices (points) connected by non-intersecting arcs (curves, lines that connect the vertices).

Definition 2: A vertex is called odd if it has an odd number of arcs leading to it, otherwise it is an even vertex.

Definition 3: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has two or less odd vertices, it has at least one Euler path.

Try to draw networks which satisfy or not the theorem above. Then don't be lazy - send me some nice solutions so that they can be included here!

 

   

See about some mathematicians who have contributed to the development of topology:

Euler (1707-1783)

Möbius (1790-1868)

Poincaré (1854-1912)

Hilbert (1862-1943)

Click here to learn more about topology.

Download a worksheet on Möbius strip by clicking on the number man.

 

Go to the Cyber Atlas, to see some topological maps of the Internet.

 

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