THE PROBLEM OF THE SEVEN BRIDGES OF KÖNIGSBERGChristopher Merck12/8/2015

LEONHARD EULER

Born in 1707 in Switzerland One of the world’s most prominent mathematicians and

physicists Died in1783 in Russia Complete set of works Leonhardi Euleri Opera Omina took 100

years to compile

CITY OF KONIGSBERG, GERMANY

Founded in 1255 by Teutonic Knights on River Pregel Given to USSR after World War II Now Kaliningrad, Russia Located on Baltic Sea

WHAT IS THE PROBLEM?

Is it possible to cross all seven bridges in such a way that you only cross each bridge once?

Euler was asked to do this, and even though he found it trivial, he was intrigued

IMAGE OF THE BRIDGES

HOW TO DO IT

The choice of route is irrelevant The sequence of bridges crossed is important Put vertices for the land masses Put edges to represent bridges Try to trace your way through it

WHAT EULER SAID ABOUT THE PROBLEM

“My whole method relies on the particularly convenient way in which the crossing of a bridge can be represented. For this I use the capital letters A, B C, D, for each of the land areas separated by the river. If a traveler goes from A to B over bridge a or b, I write this as AB, where the first letter refers to the area the traveler is leaving, and the second refers to the area he arrives at after crossing the bridge. Thus, if the traveler leaves B and crosses into D over bridge f, this crossing is represented by BD, and the two crossings AB and BD combined I shall denote by the three letters ABD, where the middle letter B refers to both the area which is entered in the first crossing and to the one which is left in the second crossing.”-Leonard Euler, from bigthink.com

IMAGE OF VERTICES AND EDGES

THEOREM ONE

If a network has more than two odd vertices, it will not have an Euler circuit

THEOREM TWO

If a network has zero or two odd vertices, it has at least one Euler path

If it has two odd vertices, then its Euler path has to start at one and end at the other

EULER’S SOLUTION

August 26th, 1735-Paper entitled “Solution problematis ad geometriam situs pertinetis’

Literally means “The solution to the problem is the geometry of position”

Consisted of 21 Paragraphs

DOES THIS PASS?

In Paragraphs 13-15, Euler talks about graphing to find a solution

He determines, that due to this path failing the first theorem, it is not possible to cross all the bridges while only crossing each particular one once

IMPORTANT TERMS

Euler Circuit occurs when a path uses each edge exactly once, and ends at the same vertex as it started

Euler Path occurs when a path uses each edge exactly once, and ends at a different vertex than where it started

WHAT IS GRAPH THEORY

Graph theory is the branch of mathematics which studies how networks (graphs) can be encoded and their properties measured

Studies the relationship between lines and points No attention is paid to the position of points or lengths of the

line

EXAMPLE OF GRAPH THEORY

APPLICATIONS OF GRAPH THEORY

Flight Patterns and Flight Control Social Media Traffic Control The Internet

VARIANTS OF THE BRIDGE PROBLEM

Several variations to the problem Identifying the nodes Prominent landmarks shown (castle, inn) Traveling Salesman Problem Four Color Theorem

STATE OF CURRENT KALININGRAD BRIDGES

Only two original bridges exist Two bombed and destroyed in WWII One rebuilt in 1935 Two were replaced by highway Total of five remain

CONCLUSION

Laid foundation for graph theory, and later topology First proof of networks Locals no longer had to think about this dilemma