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Sunday, November 29, 2015

Topology and the Seven Bridges of Konigsberg

By Courtney Correa

Topology is the mathematical study of the properties that are preserved through deformations, twisting, and stretching of objects. (wolfram.com) Tearing of the objects in topology are not allowed, and an example of two shapes that are topologically equivalent are a circle and an ellipse.   Topologists study the shapes and their properties that remain the same when stretched or compressed.  Leonhard Euler is credited with the discovery of topology of networks and in 1735 his work in this field was inspired by the Seven Bridges of Konigsberg problem.  In Konigsberg, which is now modern day Kaliningrad, Russia, a river ran through the city and created a center island.  After the island, the river split into two parts and the people of Konigsberg built seven bridges for people to use to get around the city. (mathforum.org)

Euler presented the question of if it was possible to cross all seven bridges one time, and be able to access each area of land beside each bridge.  He concluded that this is not possible, and he analyzed the geometric position of each bridge and piece of land to prove this.  With a point or node representing each piece of land, and an edge representing each route to a bridge, he counted four nodes and seven edges.  Euler looked at how many vertices each node had and noticed that the nodes could be odd or even.  An odd node has an odd number of vertices and an even node has an even node of vertices.  The problem with the seven bridges was that all of the nodes were odd, and this meant that it did not matter where you started or finished, you would end up getting stuck and not crossing one of the bridges.
When analyzing the Seven Bridges of Konigsberg, Euler created what is known today as a topology of networks.  (mathforum.org) A network represents points and lines that when connected, they create odd or even vertices.  A network can be considered traversable if one can use a pencil to trace the shape without lifting up the pencil and without going over a side more than once.  Euler’s creation of the graph of the seven bridges is not traversable because the four vertices are odd.  Euler’s graphical representation of the Seven Bridges of Konigsberg hinted at the discovery of topology because of the fact that the most important components of the problem were the number of bridges and their endpoints, not their exact positions.  Topology is not concerned with the shape of objects, and the seven bridges problem is a great example of this because we are not focusing on the actual layout, we are focusing on the graph.  

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