Who Makes Acoustic Solutions Tv

Königsberg was a popular and wealthy city of East Prussia, today his name is Pertec Kaliningrad and Russia, is located on the shores of the Baltic Sea and about 50 km from the border with Poland.

In this city two rivers meet, forming an island at the confluence. Seven bridges joined (and no, because the city was partially destroyed during World War II) the different parts of the city, as shown on the map at the time. In the eighteenth century became popular as a riddle / hobby whether it was possible to cross the seven bridges of the city, passing only once each.

Königsberg in Euler time

Kaliningrad, currently

This city is also known for being the birthplace of philosopher Immanuel Kant (1724 - 1804), but in the History of Mathematics is famous for the provision of the bridges that gave rise to this game, just at the time of Kant, which attracted the attention of the most famous mathematicians of the time.

This problem, of course, can be solved by an exhaustive study of all possible itineriarios. But in mathematics we are interested in generalizing the problem and find a simple and valid for all possible maps of cities, and even more general objects.

In 1736, the great Swiss mathematician who lived in St. Petersburg, Leonhard Euler published his "Solutio ad geometriam problematis pertinentis situs" , an article that solved the problem in the general case. This work is regarded as the birth of graph theory, used today in countless applications, and also one of the first appearances of a "new geometry" in the others only the structural properties of an object and not measurements. That is referred to the words "geometry situs" in the title Euler, words that are translated today as topology.

The first observation is that the problem can be reformulated as follows. Since The following "graph" with four vertices and seven edges, can cross it without whole spend twice for the same edge?. In the graph, the four corners represent the four parts into which the rivers divide the city, and edges represent the seven bridges. Put another way, which is more familiar to the amateurs of pastimes: Is it possible to make the drawing of the graph without lifting the pencil from the paper and not passing twice on the same edge? (You can spend twice for the same vertex).


Euler's answer is extremely simple. Suppose indeed it is possible to do the drawing without lifting the pencil from the paper. In making the drawing, we go through each intermediate vertex by an edge enter and come out the other. In particular, the number of edges that meet at each vertex of the graph, except perhaps the initial and final vertices of the drawing, must be even. If we call "valence" of each vertex the number of edges that flow into it, the above means that the problem has a solution is necessary that the graph has at most two vertices of valence par. In the case of the Königsberg graph all four vertices have odd valence, so the problem has no solution . QED