Saturday, April 13, 2013

Bridging Königsberg

The problem is practical enough: could one take an evening stroll through bygone Königsberg and cross each of the city's seven bridges exactly once, ideally returning to one's point of origin? It's easy to imagine the city's carefree flâneurs on their sunset walks along the river idly musing on the existence of the perfect path, one requiring neither retracing their way nor missing a single bridge. I have to admit, in my own wanderings I have longed for such a route, but alas, today, as then, the ideal walk remains an elusive dream for those who stroll along the water.

Leonhard Euler, the Swiss mathematician who was living and teaching in St Petersburg at the time, presumably first heard about the dilemma of Königsberg's pedestrians around 1736 from the Mayor of Danzig. Euler initially dismissed the problem has having no relationship to mathematics, but solved it nonetheless -- and in doing so, founded the new concept of graph theory and laid the first stones for topography.1

The original seven bridges of Königsberg, as they stood during Euler's life.

Future contributors to graph theory would eventually extend Euler's conclusions into a method of rendering such geographic problems through nodes and lines. Although Euler himself is often incorrectly credited with this innovation, his proof did simplify the problem into an analysis of the number of connections to each land mass. His ultimate conclusions are intuitive enough that even I can see their logic. Essentially:

If there are more than two land masses (vertices) with an odd number of bridges, then a continuous route across all bridges is impossible. (This was the case in Königsberg, since both islands and both banks of the river were connected by an odd number of bridges.)

If there are exactly two land masses with an odd number of bridges, then a continuous route is possible only if it begins on either of these land masses. (This is the case today, which makes such a route possible, but impractical, since it must either begin or end on the island.)

And if there are no land masses with an odd number of bridges, then a continuous route is possible from any location.2

The Königsberg bridge problem rendered as an Eulerian graph, in
which each land mass becomes a dot (vertex, or node), and each bridge a
line (an edge). Source.

This is all very well if you're a graph theorist, but it has little relevance to me beyond foiling my dream of a perfect walk. More interesting to me is the fate of the bridges themselves, which, like that of the rest of Königsberg, is not a clear or happy one. Of the seven bridges from Euler's time, five survived the war (although one had been destroyed and rebuilt in the late 1930s, so it doesn't count), and three of those survived the Soviet Union. The two that make up the difference were destroyed to make way for the construction of a modern overpass over Kant's Island in 1972.

Locations of original seven bridges (1-7) and more recent
additions (8-10), descriptions as follow.

1: Merchants' Bridge (Krämerbrüke, Лавочный мост)
Made of wood and built in 1286, the bridge connected the northern bank's Altstadt district with the Kneiphof on the Island. It was repaired in 1900 (when presumably it was converted into a drawbridge), survived the war, and was demolished in 1972 to allow the enormous new overpass bridge (Эстакадный мост) to be built directly overhead. Today, all that remains is a small sheared-off part of the embankment under the overpass, where it's barely possible to imagine the street once passing over the river.

The Krämerbrüke, looking towards the castle
down Kantstraße. Source.3
Krämerbrüke, opening for  a ship up the river, 1921. Source.
Standing on the island -- the two bulgy cement curves on either
side of the river are all that remain of the Merchant's Bridge. Above,
the overpass for which they were destroyed.
The overpass above Kant's Island, looking southward.

2: Green Bridge (Grüne Brüke, Зеленый мост)
Built in 1322, this bridge was converted into a moveable drawbridge in 1907, connecting the Kneiphof to the southern bank next to the stock exchange. It too survived the war, but met an identical fate to the Merchants' Bridge. If you didn't know a bridge used to stand there, you'd never be able to tell.

Before its 1907 facelift, the Green Bridge and the view from the south
bank over the island and up towards the castle in the distance. Source.
After its reconstruction, view of the bridge and stock exchange, 1908.
Under the overpass towards the stock exchange, looking directly
over at where the Green Bridge used to connect.

3: Blacksmith's Bridge (Schmiedebrüke, Кузнечный мост)
Parallel to the Krämerbrüke, this bridge was built in 1379, and restored in 1896. It did not survive the war, and now all you can see are its broken foundations on both banks of the river.

Schmiedebrüke and bridgehouse, from the island towards the northern
bank. Source.
Similar view today, from the island towards the northern bank
and the House of Soviets.

4: Butcher's Bridge (Köttelbrüke, Мясницкий мост)
Parallel to the Grüne Brüke, this bridge reached from the island to the south bank on the other side of the stock exchange. Built in 1377 and rebuilt in 1886, it did not survive the war either.

Köttelbrüke, probably from the top of the stock exchange, looking
towards the cathedral on the island. Source.
Empty foundations, from the island towards the stock exchange,
which is partially visible on the right.

5: Honey Bridge (Honigbrüke, Медовый мост)
Some maps label this as the Cathedral Bridge (Dombrüke), which makes sense since it connects the Kneiphof by the Cathedral to the large island, across from where the New Synagogue used to stand. Supposedly, the name comes from the barrels of honey with which a noble bribed the city council in exchange for the bridge's construction in 1542. It is the only original bridge on from the Island that still exists today, and the railings are heavily encrusted in marriage locks.

Honigbrüke and the New Synagogue. Source.
Drawbridge mechanism of the Honey Bridge, from the island. A favorite
bridge of tourists and brides alike.

6: Wooden Bridge (Holzbrüke, Деревянный мост)
Built in 1404, this bridge connected the north bank with the large island. It still exists and is in use today, although I wouldn’t bet money that it's one of the safer bridges in the city. One of the few remaining tram routes still rattles over this bridge.

View from the north bank, over the river towards Kneiphof. The cathedral
is visible behind the Albertina University building, which no longer exists.
Holzbrüke from the large island northwards. Source.
From the island northwards.
Taken from Kant's Island, a tram rattles over the bridge. The Synagogue
would have been to the right.

7: High Bridge (Hohe Brüke, Высокий мост)
The High Bridge, built in 1520, was the last of Königsberg's original seven bridges. In the 1880s it was converted to a drawbridge, and a brick bridgehouse was built to house the mechanical. In 1937, the old bridge was demolished, leaving only the stone pilings in the river, and a new one was constructed right next to it, which still exists and bears traffic today. The tiny bridgehouse also still exists, looking lonely and overgrown on the south bank.

View from north to south down the original Hohe Brüke, with its
bridgehouse to the left, ca. 1908. Source.
From the other side of the river, the bridgehouse, original pilings
of the first bridge, and the "new" bridge to the left.

8: Emperor's Bridge (Kaiserbrüke, Императорский мост)
This bridge is sometimes confused for one of the original seven bridges, since it's closer on the map to the others than the High Bridge. In fact, it wasn't built until 1905 -- considerably past Euler's expiration date. The bridge met its end in the war, but was rebuilt to resemble the original in 2005, as part of the fishing village project and the city's 750th anniversary. It is also popular for weddings.

Kaiserbrüke and bridgehouse, from the main shore towards the island.
The reconstructed bridgehouse looks too new and gimmicky, but
actually they did a very nice job on the bridge itself.

9 and 10: Second Overpass Bridge (Второй эстакадный мост)
Technically, this bridge shouldn't be on the list, since it had no pre-war analogue and was built recently, finally opening in 2011. But since we're talking about Kaliningrad's major bridges, it's worth mentioning. Aside from the first overpass bridge, which spans Kant's Island, this is the other main vehicular route between the southern and northern halves of the city. However, it's certainly not pedestrian-friendly, and shouldn't be counted in any bridge walking-tour.

The second overpass bridge, which serves a great deal of the north/south
traffic, and connects with Moskovsky Prospekt.

I'm not a mathematician; I'm a historian and a writer. Euler saw the bridges of Königsberg and looked for a mathematical pattern. I look for their poetic one. I've known about the Bridges Problem since before I arrived here, and I've been crossing those bridges for the last seven months, searching for their metaphor. My instincts tell me there is one, and I thought if I mulled it over and crossed those damn bridges enough times, it would eventually come to me. But I haven't found it, and I've learned that I'm not going to.

The bridges don't need a metaphor because they speak for themselves, and require no values from me. They existed, and then they didn't, like so much else. If it seems that their fate describes the city's own, well, far be it from me to pass judgments on that. From photographs, I believe that this was a beautiful city, and from its academic publications, I believe that it was a vibrant part of the Enlightenment's development of rational scientific thought. Looking out my window today, I see aging concrete and haphazard new constructions, made with little regard for the eye and less for the brick foundations they might rest upon. It is too easy to mourn for this city's transformation from Königsberg to Kaliningrad as taking all beauty and value with it -- because when comparing photographs to the views today, that's what it can seem like.

But I did not live here then as I live here now, and in two months I will be gone altogether. It is not my place to lament Königsberg, nor to condemn Kaliningrad -- and in fact I find it dangerous to think so. Raising Königsberg on a pedestal is a blind nostalgia, made doubly so by the schism between its current and past inhabitants. I cannot and will not say that, had I lived in this city in 1470, 1525, 1701, 1871, 1918, 1938, or any other period in its long and convoluted history, I would have had less censure for it. The past is a foreign country, and the future unknowably far.

Much of beauty was destroyed in the war, yes, and much more after that. But the value of a place is measured in more than just architecture, and while there are half a million people who still call this city home, then there is still value here. It is hidden sometimes, behind the concrete and the billboards, but it lives in the minds of people, not in the buildings that house them. A bridge is just a bridge, and its importance lies in the thoughts and creativity and mathematical theorems it can inspire in the people who cross and consider it.

1. Brian Hopkins and Robin Wilson, "The Truth about Königsberg," College Mathematics Journal 35 (2004): 198-207.
2. Ibid, 206.
3. The German-language Wikipedia article Königsberger Brüken has many more photos of the city's bridges, if you're interested.

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