Unique discoveries from strange questions: Euler, Königsberg's bridge and graph theory
Beginning:
e = 2.718218…
We are all familiar with this e from secondary mathematics. ln e = 1 or dy/dx(e^x) = e^x etc. All the strange sources about e have made many of us wonder about its greatness. This e is Euler's number. Mathematician Leonard Euler (Leonhard Euler) whose proponent.
This eighteenth-century mathematician introduced many symbols in mathematics, including π for pi, i for √-1, and Σ (sigma) for sum. He was one of the famous pioneers of theoretical mathematics. He has published over 500 research papers. Euler also worked on number theory, integral calculus, mechanics, optics, astronomy, fluid dynamics. One of his works was the Königsberg Bridge Problem. There is an interesting story behind its solution and the discovery of graph theory.
Euler's number
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Euler's number; Image Source: cantorsparadise.com |
Euler was born in Basel, Switzerland in 1707. During his student life he came close to another famous mathematician, Johann Bernoulli. Euler served at the St. Petersburg Academy of Sciences and the Berlin Academy at his career. While in St. Petersburg, he received a letter from the German city of Königsberg regarding a problem with the Königsberg Bridge. The city of Königsberg is surrounded by the river Pregel. The river divides the city into four parts, consisting of two main blocks and two islands. All parts were connected by 7 separate bridges. The question arises in the leisure mind of the city dwellers that if one follows any route starting from any one place, one can cross all 7 bridges without going to any bridge more than once? Look at the diagram below, the problem is explained with the help of figure-
Königsberg Bridge Problem
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Königsberg Bridge Problem; Image Source: medium.com |
The main land is divided into four parts A, B, C and D. The sections are connected by 7 bridges. Starting from anywhere A, B, C and D, is it possible to complete the journey by crossing all the seven bridges only once (without swimming of course)? Readers, look at the image and try to solve this puzzle from almost 300 years ago.
Sweat is supposed to run off. On the face of it, this puzzle seems solvable, but actually there is no solution. It is not possible to cross 7 bridges by traveling over each bridge only once. Euler did not give much importance to the problem at first. He said, it has nothing to do with mathematics. He later solved this general problem and from it Euler introduced the mathematical geometry of position, which later became known as graph theory. This theory does not actually explain what a data chart, bar or line graph means. Graph theory deals with many points or networks of points.
To explain this, Euler considered each land (mainland and island) as a point or node (node) or vertex. The connecting lines or bridges of any two bumps are lines or edges. Euler presented the problem as follows:
Euler's graph representation
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Representation of Euler's graph; Image Source: medium.com |
Euler showed that if all but the first and last nodes are connected to an even number of edges, then a journey is possible by traversing all edges only once. Such a route is called Eulerian path. On the way one is entering the territory through a bridge, and exiting through another. Thus, if bridges are connected to a terrain in pairs, then it is possible to enter and exit a terrain by using each bridge once. The number of edges (bridges) to which a node (land) is connected is called the degree of the node. According to Euler, a journey through each such edge only once i.e. Eulerian path is possible, if either of these 2 assumptions is true-
Only nodes with 2 odd degrees (start and end), rest must have even degrees.
The degrees of all nodes are even (then the start and end points are the same, such a loop is called an Eulerian circuit).
In Königsberg these conditions are not met, so this solution is not possible. Each of the four terrains A, B, C and D has an odd number of bridges (5, 3, 3 and 3). But if there were a total of 8 bridges plus one more bridge, then it would be possible to cross each bridge once to arrive at the starting point, or an Eulerian path would be formed.
Graph theory has subsequently been used in various fields. It generally deals with the relationship between different data or objects. This theory is useful in solving various layout, network, optimization, matching and operational problems. This theory is useful for finding the shortest path between two places on Google Maps. Graph theory is an important tool in the analysis of algorithms in computer science.
To find the shortest path in a network graph theory is also used. The use of graph theory in circuit design and solution in electrical engineering is undeniable. Also, graph theory has applications in various areas such as solving various statistical problems, interpreting three-dimensional images of complex simulations, analyzing molecular structures, solving minimization problems, etc.
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Leonard Euler; Image: Public Domain |
Königsberg, the city that pioneered a new era of mathematics, is no longer on the map today. The city was severely damaged by Allied bombing during World War II. After the war the area came under Russian control and was renamed Kaliningrad. Along with the history of graph theory, the old name of the city has gone down in history.
Euler spent his entire life in various researches. He published many researches with the help of others even when he was blind in his late life. According to tradition, this scientist breathed his last in Russia in 1783. In his honor, the Institute of Combinatorics and its Applications (ICA) awards the 'Euler Medal' for research. An asteroid discovered in 1973 was named '2002 Euler'.
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Euler Medal; Image Credit: sites.math.rutgers.edu |
In the flow of a traditional event, sometimes a new theory is suddenly discovered, a new horizon emerges. The puzzle of crossing the seven bridges only once, discussed in a coffeehouse by the townspeople of Königsberg, gave rise to theoretical lines of mathematics such as graph theory, which are still used to solve various problems in human civilization today.