Solving The Unsolvable – The Seven Bridges of Konigsberg: Euler’s Path To Genius

Math was never my strong suit. Basic Algebra was the limit of my arithmetical competence. Everything beyond that was a struggle. In geometry I struggled to a grade of C, Algebra II a grade of D and I dropped Trigonometry after one week. I knew getting through any university level math courses would be a struggle. Imagine my surprise then, after I got to college and found a course called Infinite Math. It was not nearly as daunting as its name. The course consisted of Maths that could be applied to the real world. My favorite of these was something called Euler Circuits, which meant finding the most efficient route for a journey. There was also the less rigorous Euler Path. Trying to figure out the most efficient Euler Circuit or Path became one of my favorite mathematical exercises.  As for the name Euler, it never meant much to me until I recognized it again after many years while reading about Konigsberg, the old capital of Prussia and today the city of Kaliningrad, in the Russian oblast of the same name.

The Seven Bridges of Konigsberg

The Seven Bridges of Konigsberg

A Problem Without A Solution – Explaining The Impossible
The Old Town of Konigsberg stood on both sides of the Pregel River. Uniquely, as the river flowed through the city it wove its way around two islands. The more famous of the two was the Kneipfhof which had five bridges going across arms of the river. Another two bridges crossed branches of the river from another island, Lomse. These seven bridges were the genesis of a puzzle that many in the town tried to solve. As one resident of Konigsberg related in a letter to Swiss mathematician Leonhard Euler, couples in the town liked to try and figure out a route to cross every bridge once without ever having to re-cross any of the same bridges again. An even tougher problem would be to do this while ending up back in the same place they began. In 1736 Euler set himself the task of proving that a solution to this problem was impossible. He did this by focusing only on the land masses and bridges. He made each land mass a “point” or in modern parlance a “node”. Each connecting bridge was an “arc”. This abstraction could then be drawn as a graph. Euler’s proof was published in 1741, six years after he first began to study the bridges problem.

The essence of the problem was how to draw this upon a sheet of paper without retracing any line or lifting a pencil off the paper. This laid the basis for the first ever theorem of graph theory. Euler’s name was given to among other things, Euler Paths which is a continuous route that passes every edge once and only once. His name was also given to Euler Circuits, a path beginning and ending at the same starting point without retracing any part of the route. Many people in Konigsberg understood from experience that there was no route that could be followed to cross all Seven Bridges of Konigsberg once and only once without retracing some part of the route. Euler’s innovation was that he could explain the impossibility of a solution and used it to develop the basis for graphs, networks and topology. Euler’s mathematical genius extended to the counter-intuitive. With the Seven Bridges of Konigsberg he proved the rationale, reasoning and intellectual uses of a problem that could never be solved.

The Wooden Bridge - in 1930's Konigsberg

The Wooden Bridge – in 1930’s Konigsberg (Credit: Eigenes Werk)

Bridging A Divide – From Konigsberg To Kaliningrad
Crossing the Seven Bridges of Konigsberg as Euler knew them is impossible today, but not because of any mathematical problems. The difficulty arises from the fact that, like almost all of old Konigsberg, most of the bridges no longer exist in their original form. Two of the bridges – Blacksmith’s Bridge and Giblet’s Bridge – were destroyed in the British bombing of the city. Both of those bridges led to and from Kant Island. The bombing which took place on two nights in late August of 1944, also leveled much of the castle and cathedral, though the latter has been rebuilt. Two other bridges – the Shopkeeper and Green Bridge – disappeared after the war to make way for Leninsky Prospekt in what had suddenly become Kaliningrad, a closed Soviet city. Thus, the Seven Bridges of Konigsberg were now three bridges in Kaliningrad. The most popular of the three that still exists today is also the only one that goes to Kneiphof, the aptly named Honey Bridge. Like many other famous bridges in European metropolises it sports hundreds of padlocks which are symbols of those romantic couples hoping these symbols will secure their love forever.

The Honey Bridge leads between the reconstructed Cathedral and the Fishing Village, two of the most famous spots in the Old Town which only adds to the foot traffic. Another bridge which is original, the Wooden Bridge, was lucky enough to escape destruction by either bombs or Bolsheviks. For historical harmony, it would be nice if all the bridges were rebuilt, only one holds that honor and it was rebuilt before the war by Germans, not afterwards by the Soviets. It is known simply as the High Bridge. The upshot of all this bridge building, crossing and destruction is that only two of the seven bridges that existed during Euler’s time can still be found in their original form today. It is easy enough to cross those two bridges without having to retrace one’s footsteps. Yet there were and still are many more bridges in Konigsberg to cross, perhaps not as famous, but just as important in their own way.

Leonhard Euler - mathematical genius

Leonhard Euler – mathematical genius (Credit: Jakob Emanuel Handmann)

Fits Of Mathematical Imagination – The Seven Bridges Of Kaliningrad
One can only speculate as to all the different problems and solutions Euler could have concocted by adding or subtracting these bridges in his theoretical fits of mathematical imagination. Today Euler would have the option of adding the newly built or refurbished Flyover and Jubilee Bridges to his equation. This has brought the total of bridges in the city back up to seven. Many things have changed in Kaliningrad and Konigsberg is no more, but the Seven Bridges problem still exists, albeit in extremely modified form.


Revenge of the Prussians – The House of Soviets In Kaliningrad (Part 2)

In 1968 the leader of the Soviet Union, Leonid Brezhnev issued a directive that the ruins of Königsberg Castle were to be blown up and cleared from the center of Kaliningrad. Brezhnev, as a true communist ideologue, stated that the castle represented “a hornet’s nest of militarism and fascism.” Despite the protests of intellectuals and preservationists, it was not long after Brezhnev’s decree that the castle’s ruins were dynamited and bulldozed. Over 600 years of history exploded into dust and then was swept from the surface of the city center. The physical remnants of the castle were gone. Beneath the surface though, the situation was much different. Subterranean chambers of the castle still existed. These would exact a bit of poetic vengeance on the Soviets. For the rest of history, the castle has been more than a memory. It has been a curse upon Soviet efforts to recreate the site in their own image.

The final remains of Königsberg Castle being demolished in 1968

The final remains of Königsberg Castle being demolished in 1968

Sinking Into Dystopia – Model for a Model City
Kaliningrad – formerly Königsberg, the capital of East Prussia – was supposed to become a model communist city where all traces of its ethnic German past would vanish. Instead the newborn city would be transformed into a place where many of its citizens wished for a return to the city’s Germanic past, at least in an architectural sense. Soon after the final destruction of the castle ruins, the Soviets set about recreating the site. They constructed a building that would come to represent them in ways they could not possibly have imagined. The plan was for a 28-story building, created from that all-time favorite Soviet building material, concrete. The building was to act as Communist Party headquarters for Kaliningrad and the surrounding region. It would be called the House of Soviets, a lasting centerpiece for the totally Sovietized city.

The problems started early and often. The site did not have the best foundations, partly due to all the destruction that Soviet demolition efforts had wrought upon it. The building was placed in an area that had been one of the castle’s moats. To make matters worse, originally the castle had been built atop what had once been marshland. Former marshland, on a former moat, turned out to be a good place for one thing, sinking. The extreme weight of the building from tons and tons of concrete served to further destabilize the site. Subsurface chambers from the castle began to collapse adding to the porous quality of the site. The sinking became known as the “Revenge of the Prussians.” The end result was nothing less than a monstrosity. Among innumerable problems were structural deficiencies exacerbated by a flawed design. The construction took longer than expected. Only 21 of the 28 stories were finished, meanwhile the interior was still uninhabitable. When the project dragged on for decades the city administration lost interest. Lack of funding finally brought construction to a halt.

House of Soviets - with a fresh coat of paint

House of Soviets – with a fresh coat of paint

Facing up to the Faceless –Post-Cold War Style
The House of Soviets was never completed as originally designed. The result was more like a house of never ending horrors, from structural to financial to aesthetic. Kaliningrad was left with an eyesore in one of its most prominent public spaces. It could be seen throughout the city, always lurking as a reminder of the failure of centralized state planning.  Utopia could not be created, but dystopia was certainly within reach. Kaliningrad never came close to the grand designs placed upon it. The House of Soviets was a microcosm of the city, impersonal, dehumanizing and unsightly. After the Soviet Union collapsed a conversation started among the city’s leadership on how to improve the look and feel of the city. Not surprisingly their focus turned to the House of Soviets. How could it not, the building never had gone away. Stolid and unyielding the building stood as a testament to waste and stupidity.

In 2005 none other than the President of Russia, Vladimir Putin visited Kaliningrad. In preparation the city decided that something must be done to spruce up the House of Soviets. New windows were installed, quite a feat that the communists had been unable to achieve. A face lift was also in order, but only in a cosmetic sense. The exterior was painted blue and white to help cover up the concrete. A paltry attempt to give the House of Soviets something it never really had: a veneer of aesthetic respectability. Whatever Putin thought of this, if he even noticed, was not forthcoming. His actions on behalf of Kaliningrad spoke louder than any paint job ever would. Putin paid for a new organ to be placed in the reconstructed Königsberg Cathedral, that Gothic symbol of Teutonic creativity and style.  Even for a man who had publicly mourned the passing of the Soviet Union, the House of Soviets was not worth Putin’s attention or considerable financial resources. Perhaps that is because the building exposes the failure of Soviet style administration.

For what they dream of - Königsberg before World War II

For what they dream of – Königsberg before World War II

Backwards Into the Future – The House of Königsberg
Since the Soviet Union’s collapse one question keeps arising: What to do with the House of Soviets? A renovation seems beyond the realm of possibility. It would actually cost less to build an entirely new structure than try to modify the existing one. With this idea in mind, some have went a step further, traveling backward into the future where they re-imagine Königsberg Castle once again taking shape, albeit in a new form. There have been plans to rebuild the castle with modern additions that could be used for commercial enterprises. There is precedent for a successful historical reconstruction in the city. One has to look no further than the Cathedral, which was rebuilt with the help of donations from ethnic Germans. Yet the Cathedral’s core still existed when its rebuilding began, a new castle would have to start from scratch. There are also questions concerning foundations at the site. Another problem is that it has long been rumored that the House of Soviets cannot be destroyed because it is owned by a mysterious figure who will not allow that to happen. If that is true then there is only one question that really needs to be asked: Where is Brezhnev when they need him?

Click here to read From Königsberg to Kaliningrad – Burying Prussia’s Past In Concrete (Part One)

From Königsberg to Kaliningrad – Burying Prussia’s Past In Concrete (Part 1)

Kaliningrad is one of Europe’s more bizarre geo-political entities. This Russian exclave, wedged between Poland and Lithuania, was created from German territory. Formerly, it was known as the province of East Prussia, taken by the Soviet Union in the last months of World War II. Today, Kaliningrad is both a city and an oblast. The latter is roughly analogous to a state or province, while the city itself occupies what up until 1946 was the historic capital of Prussia, Königsberg. Prior to World War II it was a prosperous city on the frontier of the far reaches of Eastern Europe, home to no less than thirty four churches, including a stunning cathedral that was a Gothic masterpiece par excellence. The city also included impressive fortress walls replete with bastions, neighborhood upon neighborhood of tidy and colorful homes, plus a beautiful castle at the very heart of the city. The latter structure had been the site of multiple coronations for Prussian kings. The Pregel River acted as a languid, watery thread weaving past gothic, baroque and neo-classical architectural offerings. It is hard to envision just how fantastically ominous a scene the city must have been when set beneath the leaden grey skies of a Baltic winter.

Engraving of historic Königsberg

Engraving of historic Königsberg

On The Border – Between Luck & Fate: The History of Königsberg
Such aesthetics had been preserved as much by luck, as by the hands of man. Königsberg had somehow been lucky enough to escape the ravages of war since its founding in 1255. Historically, the city was something of a safe haven. When Swedes rampaged across Eastern Germany during the Thirty Years War, the powerful Hohenzollern rulers sought refuge in Königsberg. During the Seven Years War’ when Imperial Russian troops gained control of the city, the citizens wisely bowed to their rule and saved Königsberg from the usual excesses committed by an occupying army. As Napoleon extended his might across German lands, Prussian King William III led his court to safety in Königsberg. The city was occupied again in 1807, this time by the French. There followed a time of suffering, but not destruction. The closest Königsberg came to cataclysm was brought about not by guns, but by germs. A plague that visited the city during the early 18th century claimed the lives of roughly a quarter of its population. Other than that sobering incident, Königsberg was one of the few places in Europe that was a consistently safe and prosperous place to live from the Middle Ages all the way up into the modern age. It was always just remote enough to avoid the vicissitudes of war. The military might of the Teutonic Knights, than the Prussians certainly helped safeguard the city. All of this history meant nothing during the latter part of World War II as the city was changed irreparably by the all-consuming vortex of total war.

Königsberg Castle courtyard at the end of the 19th century

Königsberg Castle – courtyard at the end of the 19th century

In late summer 1944, Allied Bombers laid waste to much of the city center and its industry. Several nights of deadly incendiary bombing were an ominous warning of the apocalypse to come. In April 1945, the city was reduced to smoking ruins. A German Army in its final death throes, vainly attempted to stave off the overwhelming might of the Red Army colossus, hell bent on vengeance. The resulting siege devastated the city. Königsberg was singled out for greater suffering than the rest of Germany both at the end of the war and afterward. Prussian militarism was considered a major reason for the two catastrophic invasions of Russia that had occurred in the previous thirty years. An overriding sense that a German invasion must never be allowed to happen again underlay the destructive actions of the Red Army as they reduced Königsberg – which means “King’s Mountain” in German – to a heaping, convulsive mass of smoldering rubble. The old Königsberg ceased to exist, but it could have been rebuilt. The Soviets had another plan in mind, one which would recreate the city in their own image.

From Facelift To Facelessness– The Creation Of Kaliningrad
As the war came to an end the Soviets were really just getting started with the city. Their vengeance extended from days into decades. A plan was hatched to Sovietize the city. It meant that hundreds of thousands of ethnic Germans would have to be relocated, what would become one of the largest forced migrations in human history. Naively many citizens of Königsberg and East Prussia returned to the region at war’s end. Approximately 800,000 were shipped eastward to far flung points all across the Russian interior. They were now slaves of Stalin’s Empire. In their place came Soviet citizens, Russians, Belarussians and Ukrainians who traveled west to repopulate a city and surrounding region that in the summer of 1946 was renamed Kaliningrad. The name was taken from Mikhail Kalinin, one of Stalin’s cronies. He had been with the dictator from the early days, but over the last decade and a half of his career was largely a faceless bureaucrat. The most notable thing about his life was that Stalin had his wife tortured and exiled to Siberia, the same fate that so many of Königsberg’s citizens experienced.
Kalinin died of cancer in 1946, during the last decade of his life he was little more than an afterthought.

Königsberg Castle in ruins - photo taken in 1950

Königsberg Castle in ruins – photo taken in 1950

Once of supreme importance, Kalinin had slowly been lost to obscurity. The same fate awaited the city which would bear his name.  As an honorific, the Soviets affixed his name to the conquered capital of East Prussia, a faceless bureaucrat for what at the time was a largely faceless city. That facelessness was soon to change for Kaliiningrad as it took on a much harsher appearance. It was not just that the people of the city had changed, but also the urban environment began to undergo an ideologically infused architectural transformation. Concrete block buildings, whether administrative or residential were constructed to house Soviet transplants. These were the physical embodiment of a brutalism just as bleak and unforgiving as Stalinism itself. The very idea of Königsberg was inexorably buried beneath this rapidly rising concrete edifices. This was a new city for a new world that bore the stamp of a mind deadening ideology. No section of the city was to be spared, especially in the center.

A Monument To Ruin – Centrally Planned Decline
At the end of World War II, that monumental symbol of Prussian power and royalty, Königsberg Castle had been reduced to ruins. These ruins lay as a silent and sullen reminder for the destruction of this once great city. In the late 1960’s the Soviet leadership hatched a plan to relegate even these ruins to oblivion. On the site where Königsberg Castle had stood for centuries a plan was conceived that would inadvertently create something unforgettable, a building which would come to symbolize the folly and waste of Soviet style communism.

Click here to read: Revenge of the Prussians – The House of Soviets In Kaliningrad (Part 2)