Suzuka Circuit

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 rebuild or not to rebuild? The case of Fujiwara-kyô

While travelling along the Rhine, we've covered a few castles which have been rebuilt and repurposed, unlike many forts in Alsace (links to some of those in that post). In Japan, the rebuilding of castle keeps to house museums was common after the Second World War, though some have remained ruin sites (further links in there). But what about a whole Imperial palace - scratch that, an entire capital?

Well, let's not get too excited: Kashihara, which was the Imperial capital for just 16 years between 694 and 710, has not rebuilt the old palace. By the time excavations began in the early 20th century, this area to the East of the town was farmland, and actually locating the lost palace and city was part of the problem.

It turns out Kashihara was the first city to be planned in the Chinese style: a grand palace at its heart, and a grid of parallel and perpendicular streets around it. In 710, the capital was dismantled and moved to Nara, before moving again to Kyôto in the late 8th century - and the grid street pattern can be found there, with numbered parallels going North to South (Nijô, Sanjô, Shijô...).

Today, the site of palace buildings are marked with these arrays of pillars (photo above with Miminashi-yama in the background), and a small museum sits on the side. There don't appear to be any plans to develop further, unlike what has been done at the palace site in Nara, which we can talk about next time.

"Stick an aircraft engine in it" part 2b - the TGV

"Wait, the TGV's electric, right?", I hear you say. You're not wrong: all TGVs in commercial service since 1981 have been electric. But this is the 1972 prototype TGV, and back then, those initials stood for Turbotrain à Grande Vitesse, continuing the development cycle of trains with helicopter engines that had already been introduced on intercity services with the RTG.

This prototype would set the standard of what French high speed rail would become: articulated units of carriages between two power cars, and the distinctive, iconic orange livery I wish they would have kept around in some capacity. The train regularly ran over 300 km/h, peaking at 318 km/h in Southwestern France in December 1972. The difference, of course, is that TGV 001 was equipped with four helicopter-derived gas turbines, two in each power car. As the oil crises hit before the production TGV was properly defined, SNCF were able to redesign the project around electric power in time for the 1980s.

After 15 years of service as a test mule, the train was due to be scrapped, but fortunately the two power cars avoided that fate. Their interiors were gutted, but the cars were saved and put on display as monuments to their builders, Alsthom, at Belfort and Bischheim (North suburb of Strasbourg). UNfortunately, they've been put by the motorway of all places, at both sites, so visiting them isn't very pleasant. At least at Bischheim, there is a footpath on the bridge over the motorway and railway yard, so it's possible to take one's time and get some decent views of the machine that started it all.

Been a busy bee today, so here are some animals from the Expo 70 Commemoration Park.

Pouancé

The small town of Pouancé is on a remarkable geographical "quadruple point", on the border of four départements! A peculiarity that dates back to the Middle Ages, when it was on the battlefront between France-affiliated Anjou and the still fiercely independent Duchy of Brittany. As such, Pouancé had a castle; its Breton counterpart was at Châteaubriant, and both towns were besieged at some point during the 15th century.

With a friend from Châteaubriant, we got to witness barriers being removed as what we guessed were maintenance or renovation works on the path around the castle were finished. The castle itself only opens during the summer, but at least we got to walk all around and get some good views of the castle, through the neighbouring park.

If you're driving into Pouancé from the West, this is how you know you've arrived:

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.

Le moulin rouge du Moulin Rouge a perdu ses pales!

Was going to take a break from posting today, but we had a bit of breaking news out of Paris: the famous cabaret Moulin Rouge's red windmill has lost its sails! They fell off at around 2 in the morning apparently, cause unknown. No injuries.

Article France Bleu Paris

I failed to mention this in the original post, but Meiji-jingû is also a high point of sumo culture. Newly promoted yokozuna, the highest rank in the sport, perform their first ring entry dance there, before their first tournament at that level. This would be their first appearance with the "horizontal rope" (that's what yokozuna means), in the shimenawa style, around their waist.

That's just happened, with the 74th yokozuna making his debut.

NHK WORLD

Hoshoryu performs first ring-entering ceremony as yokozuna | NHK WORLD-JAPAN News

About 3,500 people gathered at Meiji Jingu shrine in Tokyo on Friday to see sumo yokozuna grand champion Hoshoryu perform his first ring-ent

The New Year shrine visit: Meiji-jingû

A common New Year ritual in Japan is to go to a shrine, possibly a large one, a visit known as 初詣, hatsumôde. NHK reported that Meiji-jingû in particular was very busy. Of course, I would avoid that, so here we are with a more tranquil time, closer to the Autumn festival.

Meiji-jingû was, as its name suggests, founded to enshrine the spirit of Emperor Meiji after his death. The first Emperor of the post-Edo period presided over sweeping societal reforms, such as the abolishment of classes like the samurai, as Japan re-opened to the rest of the world and sought to catch up. The Imperial attachment is symbolised by the Chrysanthemum crests on the torii.

One of the things that can be wished for at Meiji-jingû is a happy marriage and family life, particularly at this dedicated spot with two camphor trees planted in 1920, linked with sacred rope - these are called "married trees", 夫婦楠 Meoto Kusu.

Hikone Sawayama: summit & castle

It's not about what is there today, as much as it's about what was there. Sawayama was the original location of Hikone Castle, and it is quite possibly the most important castle in Japan to have been completely lost, as it was the castle of Ishida Mitsunari, the leader of the Western Army which lost the battle to unite Japan following the death of Toyotomi Hideyoshi. There are so few traces of the castle, no obvious tell-tale structures... This small altar may trace its roots back to the days of the castle, or maybe not, but this is just about it.

Sawayama Castle was thoroughly dismanted after 1600 following the defeat of Ishida, as the new lord of the area, Ii Naomasa, appointed by the victorious Tokugawa clan, relocated the castle to a smaller hill closer to Lake Biwa. Hikone Castle, which still stands today, basically recycled the materials from Sawayama, and the view of the "new" castle complex and the lake is the main draw for hikers today.

The summit offers good views of the mountains on the other side too, with the industrial complexes near Maibara, most noticeably Fujitec and their 170 m-tall elevator test tower, in the foreground.

Koishikawa Kôrakuen in the summer

This is my favourite park in the city and I've now seen it in three of four seasons, including exactly one year ago. As August in Tokyo goes, it was very muggy and overcast, and as soon as I touched the ticket, it started raining. As I hadn't entered, the person at the ticket office offered a refund, but this was my last day in Japan so if it was going to be a wet visit, so be it.

I did shelter for a bit as the rain was rather heavy, and it proved to be a shower, so it was mostly dry during the walk through the park. Well, I say "dry", but the air was horrifically humid, I was getting just as wet when it was raining than when it wasn't! And when the Sun peeked out, wow did it burn!

Through all of this, this one heron seemed to be chilling in the middle of the main pond. Heron? Hero, more like!

After the tour, I went to the Kantoku-tei tea house for some respite, a katsu meal, some tea... and a change of shirt!