To mathematicians, they are the most famous bridges in the world. Spanning the river Pregel as it flows into the Baltic Sea, the Seven Bridges of Königsberg are today virtually synonymous with the field of topology, the mathematical study of shapes.
Their immortality was assured in 1735 when Leonhard Euler, the great Swiss mathematician, became intrigued by the city’s unique geometrical configuration. In addition to settlements on the north and south banks of the river Pregel, 18th Century Königsberg also included two large islands known as Kneiphof and Lomse. These islands were connected to the mainland as well as each other by seven bridges that were central to the city’s life.
Euler wanted to know the answer to a comparatively simple question: would it be possible for a visitor to cross each and every one of Königsberg’s seven bridges once and only once? Being a mathematician and not a tourist, it didn’t matter to him whether the visitor started in the same place as she finished.
Such a path, Euler eventually concluded, would not be possible. The visitor (let’s call her Doris) would either find herself stranded on Kneiphof or else forced to angrily patrol the riverbank waiting for the authorities to build her another bridge. Needless to say, Euler refused to consider either jumping or swimming as valid options.
Euler’s proof, published by the St Petersburg Academy of Sciences in 1741, is often considered the birth of modern graph theory – the study of networked systems which has been of fundamental importance in our modern computerised age.
By reducing the city’s complex geography to a simple arrangement of edges and nodes, Euler was able to deduce certain conditions which needed to be met if Doris were to find her way round the city. Most importantly, it was clear to him that with the exception of her start and end points, she would need to enter all the other hubs as many times as she left them. Otherwise Doris would find herself permanently stuck somewhere she needed to be able to leave.
This meant that of the four nodes, at least two needed to have an even number of connections. As can be seen from the diagram below, however, the North Bank, South Bank, and Lomse each have three whereas the central island of Kneiphof has five. This means that a so-called ‘Eulerian path’ is sadly impossible.
It has always disappointed mathematicians that the great city of Königsberg should be remembered for such a dignified failure. How much better would it have been if Doris had actually been able to complete an Euler path of the city? Or even an Euler circuit – a more challenging path that requires the start and endpoints to be identical (and is therefore infinitely more helpful to tourists).
It’s all very well to blame the city, but in my opinion the real shame is that Euler didn’t live two hundred years earlier. Or even two hundred years later. Because with the exception of a 320-year gap between 1520 and 1842 and a small blip following the Second World War, the Bridges of Königsberg have always allowed for the existence of an Eulerian path.
A Bridge Too Far
Our story starts in 1286 with Doris, wimple-clad and clog-shod, walking from the North Bank to Kneiphof across the pride and joy of Königsberg, the brand spanking new Krämerbrücke or Bridge of Merchants.
After amusing herself on the island for forty years, she is finally able to visit the South Bank in 1342 thanks to the Grüne Brücke or Green Bridge, where she spends the next 55 years believing that the world is flat and avoiding the Black Plague.
In 1377 the construction of the Kottelbrücke or Butchers’ Bridge allows her to return to the island of Kneiphof, which she can thankfully leave a mere two years later via the Schmiedebrücke, the Bridge of Blacksmiths.
Then, in 1404, the centenarian Doris is at last able to explore the wonders of Lomse thanks to the Holzbrücke or Wooden Bridge. The people of Lomse are clearly held in high regard for their powers of imagination.
She and her descendants are unfortunately stuck there for the next 116 years, until the thought of missing out on the Reformation proves too much for the inhabitants of Lomse, who build the Hohe Brücke to connect them to the South Bank. The trademark Lomse wit is once more on display – who else would have thought of calling such an elevated construction the High Bridge?
Doris’s two-and-a-half centuries of blissful uninterrupted wandering are brought to an abrupt end in 1542, when the seventh of Königsberg’s famous bridges is built to connect the islands of Kneiphof and Lomse. Known as the Honigbrücke or Honey Bridge, this is the last to be built before Euler’s arrival nearly two hundred years later.
The city retains its distinctive look until 1862, when the North and South Banks are connected directly for the first time via the Railway Bridge or Eisenbahnbrücke, which finally allows Doris to realise her five-hundred year old dream of tracing an Eulerian path round the city. She has to change her starting point, of course, this time forced to begin on either of Königsberg’s two islands.
In 1905 the city is further enhanced by the construction of the Emperor’s Bridge or Kaiserbrücke – a rare burst of invention from the people of Lomse which gives them an extra route to the South Bank of the Pregel and more or less exhausts them creatively for the coming century. Doris is still able to travel freely.
Königsberg’s final bridge is then built in 1926 to reinforce the connection between the North and South Banks, known as the Reichsbahnbrücke or National Railway Bridge. Given that Lomse now has four connections and the South Bank has 6, the 900-year-old Doris can still easily trace out an Eulerian path with her motorised wheelchair and IV drips.
Disaster then strikes the city in 1944, when heavy RAF bombardment destroys four of its nine bridges and leaves two more severely damaged. 1945 Königsberg is not a Doris-friendly place at all.
It is at this time that the city also becomes incorporated into the USSR. Königsberg becomes Kaliningrad, and what was once the capital of Prussia becomes a fiercely defended Russian territory. To this day, Russia’s presence on the Baltic Sea is guaranteed by the exclave of Kaliningrad Oblast, a piece of land roughly the size of Northern Ireland neatly sandwiched between Lithuania and Poland.
The Russians are quick to patch up the city, and by 1972 had rebuilt the Eisenbahnbrücke and Reichsbahnbrücke and constructed a brand new Flyover Bridge connecting Kneiphof to both North and South banks of the river. With both banks now having four connections each, the Eulerian properties of old Königsberg have also been retained.
Although in the intervening decades the Eisenbahnbrücke has fallen out of use, the reopening of the Kaiserbrücke in 2005 as the Jubilaumsbrücke or Jubilee Bridge has kept the number of bridges in the city in line with the historic seven. Not only that, but if you visit the city and listen very carefully, the squeaking of Doris’s wheelchair can still be heard as it careers joyously on its blissfully Eulerian way.
Gilead Amit kept putting off writing a witty one-sentence biography, saying he’d cross that bridge when he came to it. He may come up with something @gileadamit.