Euler’s fundament of 1736
Add comment September 6th, 2006 Robert Bosman
‘No problem can be solved with the same
level of thinking, that created it.’ (Einstein)
In order to create a new Architecture for both Society 4.0 and Web 4.0, we need an approach that is different form anything that created the past and the present. My proposal is to start in 1736, when the Swiss physicist and mathematician Leonhard Euler solved the so called problem of the Seven Bridges of Königsberg. The city of Kaliningrad, Russia (at the time, Königsberg, Germany) is set on the Pregolya River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point.
Euler proved that it was not possible. To do so Euler rephrased the problem in terms of graph theory, by abstracting the case of Königsberg — first, by eliminating all features except the landmasses and the bridges connecting them; second, by replacing each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The resulting mathematical structure is called a graph.
Euler realized that the problem could be solved in terms of the degrees of the nodes. The degree of a node is the number of relations connecting it with other nodes. In the Königsberg bridge graph, three nodes (B, C & D) have degree 3 and one node (A) has degree 5. Euler proved that a circuit of the desired form (using each relation once, and return to the starting point) is possible if and only if there are no nodes of odd degree. Such a walk is since then called an Eulerian circuit or an Euler tour. Since the graph corresponding to Königsberg has four nodes of odd degree, it cannot have a so called ‘Eulerian circuit’.
In the history of mathematics, Euler’s solution of the Königsberg bridge problem is considered to be the first theorem of graph theory. In addition, Euler’s recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology.
Graph theory for one helps us to understand more about human society. For instance, it may be clear that todays networks maybe presented as graphs. Doing so helps us to make complex situations transparent and manageable. The two pictures beneath for example are exactly the same network; the first one is how we see networks in everydays life; the second one how we may order it using graph theory.
So, what we need in the architecture of our future, is a strong and simple way to make our society transparant. More about the way Euler’s graph theory may help us in later posts.
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