{"id":301,"date":"2022-04-29T20:44:00","date_gmt":"2022-04-29T20:44:00","guid":{"rendered":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/harini-jayaraman\/?p=301"},"modified":"2023-05-22T14:27:15","modified_gmt":"2023-05-22T14:27:15","slug":"konigsberg-bridge-problem-and-the-evolution-of-mathematics","status":"publish","type":"post","link":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/harini-jayaraman\/konigsberg-bridge-problem-and-the-evolution-of-mathematics\/","title":{"rendered":"Konigsberg bridge problem and the evolution of Mathematics"},"content":{"rendered":"\n

Well, In this blog I will discuss one of the problems that led to the development of new branches in mathematics like ‘Graph theory’ and ‘Topology’. Let’s dive into that interesting problem.<\/p>\n\n\n\n

What’s the Konigsberg problem?<\/strong><\/h2>\n\n\n\n

Konigsberg, a city you will find a hard time finding on the world map. However, it is one of the most famous cities in the world of Mathematics due to a particular quirk in its geography. The city of Konigsberg was set on both sides of the Pregel River, and included two large islands, which were connected to each other, or to the two mainland portions of the city, by seven bridges. <\/p>\n\n\n\n

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Source<\/a><\/figcaption><\/figure>\n\n\n\n

In the 17th century, people were bothered with the question, whether they could walk around all 7 bridges crossing them only once. Let’s try it out!<\/p>\n\n\n\n

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Source<\/a><\/figcaption><\/figure>\n\n\n\n

You could very well see other routes in this video<\/a> and come up with even more routes.<\/p>\n\n\n\n

No matter how many ever routes you try you have to give up at some point because Euler answered the question in the negative and proved that there could be no route in Konigsberg that used all seven bridges exactly once and laid the foundation for Graph theory. <\/p>\n\n\n\n

How was it proved?<\/strong><\/h2>\n\n\n\n

Well, the answer he came up with, did not basically exist then, what he called the ‘Geometry of position’ , now known as the ‘Graph theory’ and which led to the fundamentals of ‘Topology’. <\/p>\n\n\n\n

Euler realized during his various trials that the geography or route taken to reach the island or riverbanks does not matter. This insight led to the simplifying of map further. What matters is the 4 landmasses and the relationship between them. We could represent each of the four landmasses here named as A,B,C and D respectively as you could see in the picture below, as a single point, which we now call as Nodes<\/em><\/strong> in Graph theory.<\/p>\n\n\n\n

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Representation in Euler’s view<\/figcaption><\/figure>\n\n\n\n

We now have 4 nodes. We draw lines or edges between them to represent the bridges. <\/p>\n\n\n\n

Formally, we define a graph G = (N, E) as a pair where,<\/p>\n\n\n\n