Seven Bridges of Königsberg#
The Seven Bridges of Königsberg is a historically notable problem in mathematics that laid the foundations for graph theory and prefigured the idea of topology. Let’s delve into the details:
The City of Königsberg:
Königsberg (now Kaliningrad, Russia) was a city set on both sides of the Pregel River.
It included two large islands: Kneiphof and Lomse.
These islands were connected to each other and to the two mainland portions of the city by seven bridges.
The Problem:
The challenge was to find a walk through the city that would cross each bridge once and only once.
The rules were strict:
Solutions couldn’t involve reaching an island or mainland bank without using a bridge.
Accessing any bridge without crossing to its other end was unacceptable.
Euler’s Insight:
Leonhard Euler, a brilliant mathematician, analyzed the problem.
He realized that the choice of route inside each landmass (island or mainland) was irrelevant.
The crucial aspect was the sequence of bridges crossed.
Euler reformulated the problem in abstract terms, focusing only on the list of landmasses and the bridges connecting them.
Graph Theory Approach:
Euler replaced each landmass with an abstract “vertex” or node.
Each bridge became an abstract connection or “edge” between vertices.
The resulting mathematical structure is a graph.
The shape of the graph could be distorted without changing its essence—only the number of edges between nodes mattered.
Euler’s Solution:
Euler observed that whenever one enters a vertex by a bridge, one must leave it by a bridge (except at the endpoints of the walk).
If every bridge had been traversed exactly once, the number of bridges touching each landmass must be even.
Euler proved that the problem had no solution because the number of bridges touching each landmass was odd.
Legacy:
Euler’s negative resolution of the Seven Bridges problem laid the groundwork for graph theory.
It also prefigured the concept of topology, which studies properties preserved under continuous deformations.