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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