Some Life In The Rock Pools Of Hashigui-iwa! The Crabs In The First Picture Were Very Small, But The

Sangaku Sunday #10

On the historical front, we previously established that mathematics didn't stop during the Edo period. Accountants and engineers were still in demand, but these weren't necessarily the people who were making sangaku tablets. The problems weren't always practical, and often, the solutions were incomplete, as they didn't say how the problems were solved.

There was another type of person who used mathematics at the time: people who regarded mathematics as a field in which all possibilities should be explored. Today, these would be called researchers, but in Edo-period Japan, they probably regarded mathematics more as an art form.

As in many other art forms (Hiroshige's Okazaki from The 53 Stations of the Tôkaidô series as an example), wasan mathematics organised into schools with masters and apprentices. This would have consequences on how mathematics advanced during this time, but besides that, wasan schools were on the look-out for promising talents. In this light, sangaku appear as an illustration of particular school's abilities with solved or unsolved problems to bait potential recruits, who would prove their worth by presenting their solutions.

Speaking which, we now continue to present our solution to the "three circles in a triangle" problem.

Recall that we are looking for two expressions of the length CN.

1: Knowing that ON = b and OQ = 2*sqrt(qr), it is immediate that QN is the subtraction of the two. Moreover, CQ = r, so by using Pythagoras's theorem in the right triangle CQN, we get

2: We get a second expression by using a cascade of right triangles to reach CN "from above". Working backwards, in the right triangle CRN, we known that CR = r, but RN is unknown, and we would need it to conclude with Pythagoras's theorem. We can get RN if we know SR, given that SN = SR+RN is known by using Pythagoras's theorem in the right triangle SON, with SO = 1 and ON = b. But again, in the right triangle CRS, we do not know CS, but (counter-but!) we could get CS by using the right triangle PCS, where PC and PS are both easy to calculate. We've reached a point where we can start calculating, so let's work forward from there.

Step 1: CPS. PCQO is a rectangle, so PC = OQ and PS = SO-OP = SO-CQ = 1-r, therefore

Step 2: CRS. Knowing CR = r, we deduce

At this point, we can note that 2r-4qr = 2r(1-2q) = 2r*2p, using the first relation between p and q obtained in the first post on this problem. So SR² = 1-4pr.

Step 3a: SON. Knowing SO = 1 and ON = b, we have SN² = 1+b².

Step 3b: CRN. From SN and SR, we deduce

so, using Pythagoras's theorem one more time:

Conclusion. At the end of this lengthy (but elementary) process, we can write CN² = CN² with different expressions either side, and get the final equation for our problem:

Note that 2*(p+q) = 1, and divide by 2 to get the announced result.

To keep or not to keep? The case of the Kyoto Imperial Palace

Completing the set of former Imperial palaces that I've visited, the question regarding Kyoto was a different one, when, after the Meiji Revolution, the throne of Japan was moved to Tokyo, ending Kyoto's 1000-year tenure as Imperial capital. The answer came swiftly: keep it. After all, it had only been rebuilt in 1855, fires having destroyed the palace multiple times in the Edo period alone.

The Shishinden throne hall, with its courtyard and traditional tachibana orange and sakura cherry trees either side of its grand staircase, was used for the coronations of Emperors Taishô and Shôwa (Hirohito) in 1915 and 1928, but since the accession of Emperor Akihito in 1989, coronations have been held at the active Imperial Palace in Tokyo.

Today, the palace is preserved and can be visited for free, albeit on a fixed tour route. Only the exterior of the buildings can be accessed, still allowing good views of the Shishinden and the gardens to the East of the palace.

Strasbourg's tramway's 30th anniversary!

On this day in 1994, Strasbourg inaugurated - or rather, resurrected - its tramway network. Like many cities in France, Strasbourg had a streetcar system until the late 1950s, when it was decided that cars would take over. 30 years of worsening congestion and pollution later, the town chose a tramway, which had made a successful return in the mid 1980s in Nantes and Grenoble, over an automatic metro to revitalise its transit service.

Unlike Nantes and Grenoble, Strasbourg looked to foreign streetcar manufacturers Socimi and ABB, who designed a fully low-floor tram with generous windows. The Eurotram was at first a 33-metre vehicle (original form seen above), which quickly proved insufficient. A lengthened version, with an extra motor module and carriage, appeared in the following years.

Personally, I quite like this tram for the massive windows, the very mechanical sounds as it runs, and the fact that the warning bell is a real bell (later models have an electronic bell which... just sounds worse). A downside I have noticed, though not for me specifically, is that it has a low ceiling.

After losing out in the 90s, national constructor Alstom won the next round of contracts for Strasbourg's trams in the 2000s. The Citadis model, fully low-floor and taller than the Eurotrams, entered service in 2005. More Citadis trams arrived in 2016, with a new design that I really like, and with special adaptations to allow it to run in Germany, as the network crossed the border to Kehl in 2017, a first for a French tram operator.

Today, the network consists of 6 lines, crisscrossing the city centre and heading out into the suburbs. A 7th line is in the planning stages, due to head North towards Bischheim and Schiltigheim. Despite refurbishment, the Eurotrams won't be around forever, and new trams are on order - more from Alstom.

Châteaubriant: a regional rail hard border

Rennes and Nantes, the largest cities in the Western tip of France, could have three direct rail routes connecting them. The one via Redon has always been used in this role; a second, via Laval and Angers, was ridiculously long until the high-speed line to Laval opened and 200 km/h-capable units were adapted for use on it; and a third... has been cut off at Châteaubriant.

The irony is, the Châteaubriant route would be the shortest in distance, at just over 120 km in length, in a fairly simple North-South direction. The trouble is, the line wasn't built with the traffic between the two cities in mind, is mainly single track, and has some steep inclines (1.5% is steep for a train). As such, it has always seen modest levels of traffic, and since the 1980s, it had been falling into disrepair in parts, and become abandoned in others.

The Nantes side saw a resurrection in the early 2010s: the line was electrified and, in 2014, tram-trains began operating. We've seen a tram-train on this blog before, but this service is only technically a tram-train, as it only uses a tram line parallel to the urban tram for a few kilometres inside Nantes, and only making one stop on that stretch. The Alstom Citadis Dualis (SNCF class U 53500) units are effectively regional trains in tram clothing, and are the only tram-trains in France to have onboard toilets.

But in France, local trains are managed by the administrative regions, and Nantes to Châteaubriant is under the authority of Pays de la Loire, so the electric wires, renovated stations and new trains went no higher. Worse, the line has been cut in half by a platform link, as shown above - and a photo on this webpage seems to indicate they used to be joined. So the Rennes side, managed by Bretagne region, has remained without electric power. I remember the stark contrast between the two sides of the regional border when I visited in early 2020: to Nantes, modern infrastructure and trains; to Rennes... nothing, and tracks that were starting to be overgrown!

It turns out the line was indeed closed for repair works at the time, and in 2021, Rennes to Châteaubriant reopened, with first-generation bi-mode Bombardier AGC (B 82500) sets, which only use Diesel power on this route, operating a peak-only service.

In conclusion, the Rennes-Châteaubriant-Nantes route has always been in the shadow of the slightly longer route via Redon, as the latter provided good connections to other major towns on the South Brittany coast: Vannes, Lorient and Quimper. The Châteaubriant route was never double-tracked, and today is a striking illustration of regionalisation, with different levels of investment on either side of the border, and through service now impossible.

Your typical Breton winter weather...

11 years ago, our group of friends was celebrating New Year on the coast of South Brittany. It was a windy time, sometimes even stormy (the drive back a day or two later was possibly the most dangerous I've ever done), and it made for some impressive shots of waves breaking at Port aux Moines.

Out of service: the Trieste-Opicina tramway

While the reopening of Notre-Dame cathedral in Paris is making big news, and while I'm in a bit of a tram phase on the blogs, spare a thought for the tram line between Trieste and Opicina, closed following an accident in 2016. And it's a real pity, because it was a wacky one.

Trieste is a city by the Adriatic Sea, surrounded by steep hills - and I mean steep. Opicina is 300 m higher, and the tram line features gradients as steep as 26% - link to the Hohentwiel hike for scale. Steel wheels on rails weren't going to be enough...

Initially, the steepest section was built as a rack-and-pinion railway, but in the late 1920s, it was replaced by a funicular system. Cable tractors would be coupled to the streetcars to push them up the hill, and control their descent on the way down - that's the curious boat-like vehicle in the photos (at least I'm getting boat vibes from it). The picture below shows just how steep the climb is.

In the later years of operation, these cable tractors were remotely controlled from the tram. The streetcars themselves date back to 1935, with wooden doors and fittings, making the Trieste-Opicina tramway a charming and technically unique heritage system.

Sadly, the line is not running. Two streetcars collided in 2016, they were repaired, but service has not resumed. One vehicle, coupled to the cable tractor, remains stationary at the foot of the climb, near where the second photo was taken. A look on Google Street View shows that cars are now habitually parked on the disused tracks. The number 2 tram route between Trieste and Opicina is currently served by the number 2/ bus.

Schwarzwaldbahn Erlebnispfad

or the Black Forest Railway Adventure Trail!

The railway through Triberg climbs the hill opposite the waterfalls we covered yesterday, to get out of the Gutach valley and proceed to Villingen. But inclines are notoriously difficult for trains, as metal-on-metal contact yields little friction, so, like many other modes of transport, whether roads or even footpaths, the railway weaves its way up... but on a larger scale, as trains aren't as maneuverable!

The result is this loopy section between Hornberg and Sommerau, 11 km apart in a straight line, but the railway is 26 km long! It climbs 447 m at an average gradient of 1.7% (which, again, for a train, is hard work), and with over 30 tunnels to get through the irregular terrain. Today, an "Adventure Trail" complements the route opposite Triberg, providing hikers with amazing viewpoints and some chances to get close to the tracks.

Of course, this climb for hikers is far, far steeper! The route I took started with a strenuous 15% over 700 m. One would be very happy to find this bench after that climb - Liegewagen meaning "sleeper car"!

The trail has stations with information boards about various aspects of the railway, and, at the viewpoints, the timetable! Perhaps a bit of a downer is that there isn't a huge amount of traffic: just two DB Regio trains per hour (presumably the ones you came in on), one InterCity train at weekends, no high-speed ICEs, no freight. But the views more than make up for the low variety, and, at a decent pace, it is possible to be at a viewpoint for each passage and not miss one.

Most regional trains on this section of the Schwarzwaldbahn are push-pull sets with Dosto carriages - that's short for Doppelstockwagen, or double-deck. The end carriage seen above is a (deep breath)... DBpbzfa 766, each letter being short for some technical info allowing rail workers to know at a glance what they're dealing with: D is for double-deck, B second-class seating, p has air conditioned coach rooms (as opposed to compartments), b has wheelchair accessibility, z has a centralised electrical installation, f has a driving cab, and finally a means that the driver can operate the doors alone. 766 is the series number. These Dosto sets are usually driven by Baureihe 146 locomotives.

All in all, the Erlebnispfad can take up most of an afternoon (I completed the circuit in around 3 hours), it was a very satisfying walk and experience to be able to make the most of each vantage point. There are some other attractions along the route that I haven't mentioned - probably a short post tomorrow. For people who like hiking and trains, this trail at Triberg is worth doing!

Strasbourg's Christkindelsmärik

One of Europe's oldest Christmas market, and likely the most famous in France, is Strasbourg's. Its existence is attested as far back as 1570, appearing in the years following Protestant Reformation. Today it is a sprawling event, covering all the main squares of the central island of Strasbourg, and it's very busy, especially in the evenings and on weekends.

The traditional Alsatian name of the market is Christkindelsmärik, "the market of baby Jesus", while the city's more recent branding (since the 1990s) proclaims the town as "Capital of Christmas".

Pictures from 2018 - I haven't been to this year's market yet, but I plan to once my workload deflates - I get my annual stock of jams from the local producer's market!

The Osaka Monorail

The final post in this mini-series on the Osaka Expo 70 Commemoration Park is the obvious: how does one get there? Considering the Expo was all about bringing together the world's shared (or non-shared) visions of the future, the Monorail seems perfect! Except this line wasn't launched in time for the Expo - it was a later project, opened in 1990.

The straddle-beam monorail links Osaka Airport to Kadoma, and is due to be extended by the end of the decade. A short branch line pops out just to the East of the Expo 70 Park station, so a visit to the park is also a chance to see some mighty impressive sets of monorail points either side of the station, as the West side (above) also connects to the depot.

Palais Universitaire de Strasbourg

In 1870, a broad coalition of German nations waged a war against France. Napoleon III's Second French Empire fell swiftly, and the Second German Reich formed on the back of this victory, with a Kaiser at its head. Germany annexed Alsace and Moselle, and had particularly grand plans for Strassburg, the capital of the region. While the city centre on the island was repaired after heavy shelling, the Germans decided to develop around it. To the North, the Neustadt, "New Town", was built, with, at its core, a wide avenue with a palatial residence for the Kaiser at one end, and a University Palace at the other, the storefront of a brand new campus complex. It's still part of the university today, with classrooms and all.

Completed in 1884 in a neo-Renaissance style as a monument to Germany's newfound power, the Palace is richly decorated with statues, ranging from effigies of Germania and Argentina - no relation, the Roman settlement at Strasbourg was called Argentoratum -, and historical figures of German science and thought: below, theologist Martin Luther, mathematician Gottfried von Leibniz, astronomer Johannes Kepler and educator Johannes Sturm. Though probably not a direct response, the Eiffel Tower would provide a similar list of French greats a few years later.

France regained Alsace following World War I, and would de-germanify several buildings in the Neustadt area. The home of Germania insurances became the Gallia building, and the effigy of Germania, restored below, was removed from the University Palace. A similar process would be undertaken in 1945, after Alsace was once again be annexed by Germany in 1940 - the University Palace would need de-nazifying.

However, a few symbols remain as a reminder of the Nazi oppression. In the grand hall, called the Aula, a subtle swastika appears among the geometric patterns on the floor. It sits in a corner, as lonesome as it is loathsome. I thought I had a picture of that swastika, but looking through my folders, I don't appear to have taken photos of the interior at all. I seem to remember thinking "nah, it's weird to take pictures of Nazi swastikas". It's still weird, right?