The mystery of the Königsberg bridges – 07/26/2022 – Marcelo Viana

I like to tell this story to explain why math is so much more than “the science of numbers”, because it’s a tale of beautiful math ideas that have little or nothing to do with numbers. Ironically, the hero, the great Leonhard Euler (1707–1783), had a different opinion. But we have three centuries ahead of him.

The story begins in 1255, when the city of Königsberg was founded by the Teutonic Knights at a fork in the River Pregel. The name (“King’s Mountain” in German) was a tribute to King Ottokar II of Bohemia. A gateway to the rich trade of the Baltic Sea, Königsberg soon became one of the most prosperous Germanic cities and later the capital of the Duchy of Prussia.

At the beginning of the 18th century, the settlement stretched over four regions —the two banks of the Pregel, the island of Kneiphof, and the landmass (Lomse) between the branches of the river upstream of the bifurcation — connected by seven bridges: two bridges to Kneiphof and another to Lomse on each bank, plus a bridge linking Kneiphof to Lomse.

By this time the following problem had become folklore in the city: is it possible to take a tour of the four regions crossing each bridge exactly once? It was proposed to Euler around 1735 by the astronomer Carl Gottlieb Ehler (1685-1753), who was later mayor of the Prussian city of Danzig (modern-day Polish Gdansk).

Euler was not impressed. In the reply sent to his colleague in March 1736, he wrote: “You must understand, noble sir, that this kind of solution has little to do with mathematics. anyone, since the solution is based only on reasoning, and the discovery does not depend on any mathematical principle”.

But by then Euler had already submitted the solution for publication in the annals of the St. Petersburg Academy of Sciences (it would not be published until 1741). Far beyond solving the Königsberg problem, this work founded a new mathematical discipline, graph theory.

“Its origins were humble, even frivolous,” wrote the British Norman Briggs, “but graph theory captured the interest of mathematicians, becoming a subject with surprisingly profound results.” And with applications in numerous areas of science and technology, which move billionaire sectors of the economy.

I will continue the story next week.

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