A Few Views Of The Saar And Mosel Rivers In Germany, Which Recently Burst Their Banks Due To Heavy Rain.

Another edge of a fog bank on the bridge between the Port du Rhin and Citadelle sectors of Strasbourg. The railway is a freight line, I'm not sure if it's used much. The river, however, regularly sees barges passing by. Anyone or anything on the river had to be on the lookout that morning...

Sangaku Saturday #8

Having established that sangaku were, in part, a form of advertisement for the local mathematicians, we can look at the target demographic. Who were the mathematicians of the Edo period? What did they work on and how?

The obvious answer is that the people in the Edo period who used mathematics were the ones who needed mathematics. As far back as the time when the capital was in Kashihara, in the early 8th century, evidence of mathematical references has been uncovered (link to a Mainichi Shinbun article, with thanks to @todayintokyo for the hat tip). All kinds of government jobs - accounting, such as determining taxes, customs, or engineering... - needed some form of mathematics. Examples above: 8th-century luggage labels and coins at the Heijô-kyô Museum in Nara, and an Edo-period ruler used for surveying shown at Matsue's local history museum.

As such, reference books for practical mathematics have existed for a long time, and continued to be published to pass on knowledge to the next generation. But sangaku are different: they are problems, not handbooks.

More on that soon. Below the cut is the solution to our latest puzzle.

Recall that SON is a right triangle with SO = 1 and ON = b. These are set values, and our unknowns are the radii p, q and r of the circles with centres A, B and C. While these are unknown, we assume that this configuration is possible to get equations, which we can then solve.

1: The two circles with centres B and C are tangent to a same line, so we can just re-use the very first result from this series, so

2: Also recalling what we said in that first problem about tangent circles, we know that

Moreover, PA = AO - OP = AO - CQ = (p+2*q) - r. Thus, using Pythagoras's theorem in the right triangle APC, we get a new expression for PC:

since 2(p+q)=1 (the first relation). Equating the two expressions we now have of PC², we solve the equation for r:

again using the first relation to write 2q-1 = -2p.

It only remains to find a third equation for p to solve the problem.

More faces on Dôtonbori

There are the classics on Dôtonbori: Kuidaore Tarô, there's a well-known animatronic crab, and of course the Glico man. I reckon this ramen-loving dragon is my personal favourite.

... though, giving it some thought, I quite like the hand holding a sushi and Spiderman reaching for a pearl too.

But oh boy, are you ready for some real randomness?

This is the front of the Dôtonbori Hotel. According to Atlas Obscura, the hotel had these pillars made to symbolise them welcoming people from all over the world - the faces represent East Asia, Africa, the Middle East and Europe -, at a time when domestic tourism was dwindling (early 1990s).

Notre-Dame... de Strasbourg

I know, I know. Notre-Dame in Paris just reopened. But Notre-Dame is a very common name for churches in France. In fact, we covered one in Le Havre not that long ago, possibly one of the smallest cathedrals in the country. At the other end of the scale, one of the largest, if not still the largest, is Notre-Dame de Strasbourg. Built during the same time period as its Parisian counterpart, its facade has striking similarities: the grand rose, the two square towers at a similar height (66-69 m)... but while Paris stopped in 1345, Strasbourg kept going for almost a century, filling in the space between the towers, and adding a whopping octagonal spire on one side, reaching 142 m above ground.

Of course, there were plans to make the monumental facade symmetric, but the ground under the South tower wouldn't support the weight of 76 m of spire. In fact, huge structural repairs had to be made during the 19th century to avoid collapse.

The cathedral was the world's tallest building for a couple of centuries, from 1647 to 1874. Considering it was completed in 1439... Yeah, it didn't grow, it owed it title to the Pyramids of Giza shrinking from erosion and taller spires on other cathedrals burning down. Then it lost the title when churches in Hamburg, Rouen (another Notre-Dame Cathedral) and Köln were completed.

But talk of records is just talk, and 142 m is just a number, until you're faced with it. My favourite approach to the cathedral, to truly give it is awesome sense of scale, is the one I inadvertently took on my first proper visit to Strasbourg. From the North end of Place Gutenberg, walk along Rue des Hallebardes. The town's buildings will hide the cathedral from view for a moment, only for it to reappear suddenly at the turn of a corner, much closer, the spire truly towering over the surrounding buildings which also dwarf the viewer. I don't pass by there too often, to try to replicate the breathtaking reveal.

PS - We've already done a piece on the astronomical clock housed in the cathedral, an absolute treasure.

Something you don't see every year down South: ice! And not just a little, some proper icicles on this fountain on Place de la Trinité in Toulouse.

Plus a bit of snow sticking to the ground in the parks, as seen from the Natural History Museum.

I know that a Doge is something else in Venetian history, and I knew it back in 2015 too (though I didn't look into what it was exactly).

But at the time, the meme was in full swing, and I couldn't resist picturing this sign. I don't know if it's a good hotel though, the reviews aren't rave...

ありがとうかぼす！

Sangaku Saturday #12

Having mentioned previously how mathematical schools were organised during the Edo period in Japan, we can briefly talk about how mathematicians of the time worked. This was a time of near-perfect isolation, but some information from the outside did reach Japanese scholars via the Dutch outpost near Nagasaki. In fact, a whole field of work became known as "Dutch studies" or rangaku.

One such example was Fujioka Yûichi (藤岡雄市, a.k.a. Arisada), a surveyor from Matsue. I have only been able to find extra information on him on Kotobank: lived 1820-1850, described first as a wasanka (practitioner of Japanese mathematics), who also worked in astronomy, geography and "Dutch studies". The Matsue City History Museum displays some of the tools he would have used in his day: ruler, compass and chain, and counting sticks to perform calculations on the fly.

No doubt that those who had access to European knowledge would have seen the calculus revolution that was going on at the time. Some instances of differential and integral calculus can be found in Japan, but the theory was never formalised, owing to the secretive and clannish culture of the day.

That said, let's have a look at where our "three circles in a triangle" problem stands.

The crucial step is to solve this equation,

and I suggested that we start with a test case, setting the sizes of the triangle SON as SO = h = 4 and ON = k = 3. Therefore, simply, the square root of h is 2, and h²+k² = 16+9 = 25 = 5², and our equation is

x = 1 is an obvious solution, because 32+64 = 96 = 48+48. This means we can deduce a solution to our problem:

Hooray! We did it!

What do you mean, "six"? The triangle is 4x3, that last radius makes the third circle way larger...

Okay, looking back at how the problem was formulated, one has to admit that this is a solution: the third circle is tangent to the first two, and to two sides of the triangle SNN' - you just need to extend the side NN' to see it.

But evidently, we're not done.

Japan's first Concorso d'Eleganza

A Concorso d'Eleganza is basically a beauty pageant for historic cars. Based on similar contests for horse-drawn carriages, called Concours d'Élégance, the Italian name references the fact that the most famous competition in the genre for cars has taken place in Italy since 1929.

The Japanese version was launched in 2016, with the amusingly-named "Artistic Cars at the World Heritage, since 2016" exhibit at Nijô-jô, Kyôto. I say amusing, because they were using the "since 2016" tag in 2016, and... it hasn't used that name since! Nonetheless, the sight of these immaculately preserved cars in the lovely palace gardens was impressive.

Among the cars built between the 1950s and 1990s, there was an exhibit with rare Zagato specials. Particularly obscure is the Autech Zagato Stelvio, shown above and below. Of the three names, Zagato is the best known: they are an Italian coachbuilder, who make unique bodywork. Autech is a Japanese tuner, working most closely with Nissan. And the Stelvio is a late-80s mix of all that: a modified Nissan Leopard with very unique design choices, such as the wing mirrors in the fenders.

The Concorso d'Eleganza Kyoto is set to resume this year, having been put off due to the pandemic and the restrictions surrounding it since 2020. The principle hasn't changed: they're going to show some remarkable historic cars in a historic Japanese setting - at Nara this time.

Some amusing art in Matsumoto

In my previous post, I mentioned I was taking the train from Nagano to Tokyo, but my destination for the day was actually Matsumoto. There's quite a lot to see there, mainly the castle, but I'll get to that another time. For now, I'd just like to share some lighter, more amusing details.

This awesome statue depicts two samurai frogs riding on toadback - the poor thing seems to be overwhelmed! That would be because the street by this river has adopted the frog as a mascot. According to local history, the river used to have loads of frogs, making it a rather noisy place (I bet, if my memories of Futami, Mie, are anything to go by!). But after a typhoon caused a flood, the frogs left for higher ground and didn't go back to the river, so loads of frog statues were erected instead.

On the bridge near the entrance to this street and near the castle, other creatures could be found: "Y-cats", created by Yamazaki Takashi. I think this was a temporary exhibit.

Oh alright, I'll tease the castle...

Three churches in Le Havre

This building very much stands out in Le Havre's rebuilt city centre, as it is far more ornate than its surroundings. It's the oldest building in Le Havre, completed in 1638, just 120 years after the founding of the town. Damage by Allied bombings but not completely destroyed, the old church was kept and restored, and, with Le Havre becoming a major town again, it got its own diocese in 1974. The church then became Notre-Dame Cathedral of Le Havre, and is now approaching 50 years in the role (anniversary in December).

As it had been decided to keep the old church, Auguste Perret, the architect in charge of the reconstruction, would build a church of his own elsewhere: the new Church of Saint Joseph, a blocky concrete structure, perfectly in sync with the rest of the urban project. Where the cathedral sticks out as its rounded, classic facade contrasts with the angular buildings around it, Saint Joseph's stands out with its tall central spire. Culminating at 107 m, the tower is inspired by lighthouses, a symbol of Le Havre's maritime nature.

Somewhere in between, other churches further away from the port also survived the war - not without damage, but restoration was chosen over replacement. In the foreground, Saint Vincent de Paul's Church was built in the 19th century in a neo-Romanesque style, its central tower reminiscent of the abbey on Mont Saint Michel.