In 2017 I visited Kaliningrad, which before World War II was the German city of Königsberg. After World War II the Soviet Union took the city and its surrounding region; Germany was not exactly in a position to raise any objections. It was renamed Kaliningrad in 1946. The area around the city shares borders with Lithuania and Poland, as shown in the map below, so when the Soviet Union collapsed Russia was left with this separate connected component on the Baltic Sea. (It was also part of a separate connected component of Germany after World War I - look at a map of the Weimar Republic.) Traveling to here from Moscow by plane is considered a domestic flight even though it requires going over at least two other countries.
Königsberg is famous in mathematics as the city from the Seven Bridges of Königsberg problem. Euler's solution to the problem, in the 1730s, links him forever to Königsberg although I don't think he ever visited the city. An account of Euler's original solution is here. Below are views of the center of Königsberg, from 1641 and from 1930 (aerial photo). Both images are taken from here. The seven bridges are all visible in the first picture: five bridges connected to the island and two more on land to the east of it. (In the photo a bridge in the lower right is out of range.)
The island at the center of the bridge problem is called Kneiphof. The large building on its eastern end is the Königsberg cathedral. Inside the cathedral today there is a scale model of the region around Kneiphof from the pre-war era, shown below, and you can see they put one too many bridges on the north side.
Here is what the Kneiphof looks like today, courtesy of Google Earth.
Kneiphof and the rest of the center of Königsberg was destroyed by British bombing in 1944, about a year before the Russians arrived. What had for centuries been a highly developed island is now largely a park. The only building on Kneiphof that was rebuilt after the war was the cathedral, in the 1990s. The Russians left the cathedral in ruins for essentially 50 years after the war ended.
As the Google Earth image above shows, what had been seven bridges are now reduced essentially to five: there are still the three bridges to the east (one connected to the island and two not), and to the west there are two bridges. But really there are only four bridges, since the two bridges on the west side are really a single modern multilane bridge crossing over Kneiphof at the height of a 3-story building with pedestrian staircases down to the island. Below is what the bridge on the east side of the island looks like, with the Königsberg cathedral behind me. (Update: In front of me, behind the camera, is the former location of a synagogue in Königsberg, which the Nazis destroyed during Kristallnacht. Eighty years later, it was rebuilt and reopened in 2018.)
I was hoping somewhere on the island there would be a plaque about the bridges problem, but I did not find anything there. However elsewhere in Kaliningrad, along the Peter the Great embankment between two buildings belonging to the Museum of the World Ocean - the aquarium and the Maritime Königsberg-Kaliningrad Packgaus - there is a small building with a mural about the seven bridges problem. See the photo below, and a Google street view is here (update: if the Google street view is inaccurate, let me know; I've had to fix it at least once).
My excitement upon seeing this public monument to a math problem turned to shock when I read the description in the box at the bottom of the mural.
Here is a translation: “Widely known among the citizens of Königsberg was the puzzle of seven bridges. How can you walk across every bridge without passing over any of them more than once? Even the leading mathematician of the 18th century, Leonhard Euler, was unable to solve this problem, but it led him to a series of important discoveries. Try to solve it yourself!”. Yes, it really says Euler did not solve the problem. Fake history!
In the gift shops of Kaliningrad I found nothing about Euler and the seven bridges, but everywhere stuff is sold that mentions Immanuel Kant. Königsberg is where he spent his whole life as well as his afterlife: he is buried on Kneiphof, which nowadays is also called Kant Island. Kant's tomb, which was hardly affected by the fighting during WW II, is next to the cathedral on Kneiphof, and this apparently was a reason the cathedral was not completely removed with all the other destroyed buildings on the island after the war.
One of the last battles of World War II was the Battle of Königsberg in April, 1945. Königsberg was the easternmost large city in Germany and Hitler demanded that his army hold the city at all costs. After the general in charge there, Otto Lasch, surrendered to the Soviet Army, Hitler sentenced him to death in absentia. At the end of the month Hitler was dead and Lasch lived until 1971. The Russians converted the Nazi officer bunker in Königsberg into a museum of the battle there. Below is the entrance to the bunker, the hallway through it, and the room where Lasch was deciding to surrender.
The museum has dioramas showing battle scenes, including one near the railway terminal. The three triangular roofs in the diorama can still be seen at the terminal today (second photo, taken from here).
Gdansk in Poland is geographically close to Kaliningrad, and it too was largely destroyed during the war, but the Poles have carefully rebuilt much of the center of Gdansk to its pre-war appearance and the reconstruction continues today. Before and after photos can be seen here. (An analogous page in Russian is here.) The Russians had no cultural interest in Königsberg for many decades after the war and replaced most wrecked buildings in Kaliningrad with ugly apartment towers. Compare the relatively new building on the left below with the typical Soviet apartment building in the background.
Gdansk has a connection to the Königsberg bridge problem: scientists from this city, when it was part of Germany and called Danzig, first brought the problem to Euler's attention (see here). One of them, the astronomer Ehler, wrote to Euler seeking a solution. (Ehler later became mayor of the city. It's hard to imagine a mayor today being concerned about a math problem more complicated than figuring out if the number 0 is even.) Euler's reply to Ehler is interesting, because in it Euler wrote that he did not consider the bridge problem to be about math at all: “this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle.” (If you read Latin, the original version of Euler's letter can be seen on electronic pages 345-354 here, with the excerpt above being on p. 352 starting with “solutionem hanc” 16 lines from the bottom. If you read Russian, see pages 333-344.) Despite Euler's protests to the contrary at the time, the Königsberg bridge problem is now considered part of two areas of math: graph theory and topology. This nicely illustrates what the physicist Richard Feynman once said: “Everybody who reasons carefully about anything is making a contribution to the knowledge of what happens when you think about something. And if you abstract it away and send it to the Department of Mathematics, they put it in the books as a branch of mathematics” (see p. 45 of Feynman's Character of Physical Law or watch starting at 22:16 here).
Within sight of Kneiphof is a building belonging to Rostelekom, a major Russian phone company, on which they advertise their internet services (it says on the blue background “Number 1 Internet in Russia”) using an astronaut.
I wrote astronaut, not cosmonaut, deliberately. The person in the picture is actually an American, Bruce McCandless, making the first use of a jetpack in outer space (in 1984). The image is identified as McCandless in many places on the internet, such as here and here (as well as on the Wikipedia page about him). It's nice to see that even if the Russian government views Americans negatively, one of their state-owned companies uses an American to promote itself.
Behind the Rostelekom building is a monument either to the number e or to Internet Explorer.
Back on Kneiphof is a monument to π, sort of.
That is really a monument to artists, whose faces are along the top (and back).
Königsberg is where David Hilbert grew up and received his doctoral degree and first academic position, at the University of Königsberg in the 1880s. As a professor in Königsberg he proved the Hilbert basis theorem and began writing the Zahlbericht on algebraic number theory. In 1895 he moved to Göttingen, where he worked in analysis, mathematical physics, and the foundations of geometry. He returned to Königsberg in 1930 to give his retirement address at a meeting of the Society of German Scientists and Physicians. The day before Hilbert's address, Gödel announced at another conference also in Königsberg his first incompleteness theorem, which showed Hilbert's dream that all math problems can be proved or disproved is impossible. This is explained in a Veritasium video here. (Hilbert first appears at around 8:00, and the Gödel announcement is at 14:50.)
In the Königsberg cathedral is a memorial to “Significant Scholars at the University of Königsberg”, shown below. Near the end of the list is Hilbert and also Minkowski, who studied with Hilbert when they were both students. Just above Hilbert are the mathematicians Schoenflies and Weber, and a little higher are Jacobi and the astronomer Bessel. They were all professors there. The list is missing Ferdinand von Lindemann, who was the doctoral advisor of Hilbert and Minkowski during his 10 years there; Adolf Hurwitz, who was there for 8 years; Konrad Knopp, who was there for 7 years; and Richard Brauer, who was there for 8 years. In Königsberg, Knopp wrote his famous book on infinite series (1922) and Brauer defined the Brauer group (1929) and classified the central simple algebras over number fields with Albert, Hasse, and Noether (1932). Goldbach (who has no known image) was born in Königsberg and attended the University of Königsberg. The mathematician Jürgen Moser was a teenager in Königsberg during WW II, and the difficulties he faced during this period are described in his biography here.
I was in Kaliningrad at the end of July, which is when Russia marks Navy Day (last Sunday of the month). Since Russia's Baltic Sea fleet is based near Kaliningrad, some ships and soldiers came to Kaliningrad that day with weapons for the public to see. No signs said non-citizens were forbidden, so I went. What did I find?
Among the weapons above, one was dated from the 1950s.
Here is the high point: an American on top of a Russian tank. And nobody objected (admittedly almost nobody knew).
The lunch, typical of military meals, was kasha, bread, and tea.
This meal was not free, costing 100-200 rubles (just a couple of dollars), which disappointed soldiers there who expected to eat for free.
Brochures were available for those interested in becoming contract soldiers for the Russian naval and land forces. It turns out I am past the age limit for this career path.
The stereotype for how current and former Russian soldiers celebrate some military branch holidays is to get drunk and attack other people. One comic take on such activities is this video, which runs events in reverse: the main characters unbreak bottles on their heads, give people their clothes back, unstab car tires, and so on.
Amber is a big deal in Kaliningrad. Did you know that 90 percent of the world's amber mining is from around there? In the 1930s thousands of people were employed at Königsberg's State Amber Manufactory. More information about what they made is here. When the Nazis stole the Amber Room outside St. Petersburg they displayed it in Königsberg Castle before it disappeared at the end of the war, probably destroyed when the castle was bombed and burned down. The reconstructed Amber Room was built with amber from this region. In Kaliningrad one of the old German forts that was not destroyed during the war is now the Amber Museum, which details the history of work with amber and has some incredible sculptures made out of it. The ship in the first photo below is an amber model of the icebreaker Lenin. It is a duplicate of a gift to Eisenhower that is on display at the Eisenhower Presidential Library in Kansas. The second photo shows a violin and world map made out of amber.
Outside the museum I found a local resident with a strongly worded t-shirt (not sure if he knows what it means).
Here are a few random sightings around Kaliningrad: the strangely named light and ceiling store A priori (no apparent connection to math despite the name), a tea drink with trademark infringement written all over it, a non-human resident of the city, and an ad for those ubiquitous spinners.
On the last day in Kaliningrad I visited Svetlogorsk, which was called Rauschen when it had been part of Germany. It is on the Baltic Sea, and the sea floor drops rather slowly; even children were able to stand in the water rather far from shore. In the town was yet another model of Kneiphof and its bridges.
Many kiosks in Svetlogorsk sell amber sculptures. The top shelf of the photo below shows several well-known cartoon characters in Russia in amber form.
One store had a water fountain out front where, for 100 rubles, anyone could use a net to collect and take away their own amber pieces that the store periodically dumped into the water. Now I have my own bottle of amber, but I don't know what I can do with it.
During the train ride from Svetlogorsk back to Kaliningrad I saw some spectacular views of the sun setting.