We've Seen These Trains Before, Quite Recently In Fact, But They're Back, On A Much Brighter Day To Really

Hello! I just saw your post about the conference. I know it's very niche, but I'd love to hear / read more about your sangaku presentation. I actually went back to Konnō Hachiman-gū this afternoon, hoping to see more examples, but no such luck. (I cannot decipher them, of course, but I taught English at a faculty of engineering, and my students could. Sometimes. )

I'll put together something about the shrine, but どうぞお先に。Nudge nudge hint hint.

Hi, thanks for the message!

The presentation was in two main parts: first the historical context of the Edo period and function of sangaku in developing mathematics during that time, and second a closer look at Kashihara Miminashi Yamaguchi-jinja's example with a modern solution. I can't read the sangaku in full, but I have been able to pick out the parts with numbers and compare some of their results with the formulas.

I can probably put together a mini-series at some point. Which parts would you want to hear more about? (That's a general question btw: anyone can reply and add the conversation of course.)

Cascade de la Serva

In the woods above Neuviller-la-Roche and Natzviller are the Serva Falls. While not the largest we've covered, they are still sizeable, with the river flowing over rocks a small distance either side of the main fall.

It's a good hiking destination from Rothau or even Schirmeck station, depending on how far you're prepared to walk. I started in Rothau, had lunch by the falls, down the river back to Rothau, and wound up adding a walk in the valley to Schirmeck while waiting for the next train...

A good hike I'd like to do is 2-day, 30-km walk from Rothau up past the falls to Champ du Feu, the highest point in Bas-Rhin, and on to the plains at Barr. But I wouldn't want to do it alone.

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...

Update: Moulin Rouge has got new sails, and celebrated with some French cancan in the street.

Article France Info

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

Sangaku Saturday #4

In the previous info post, we went over the debate on the religious aspect of sangaku, and the fact that the absence of prayers on these tablets was more puzzling to some than the mathematics. As such, the tablets are not ema prayer tablets, but donations, which usually don't feature prayers on them. Case in point, some consecrated sake and French wine seen at Meiji-jingû in 2016.

Beyond wishing for good fortune and health, such donations serve two very worldly purposes: to contribute to the life and prestige of the shrine or temple (having a famous contributor makes the shrine famous by association), and to advertise the donor in return, as their name is on display. See this large torii at Fushimi Inari Taisha paid for by TV Asahi (テレビ朝日).

With that in mind, Meijizen's cynical comment from 1673 that sangaku aim "to celebrate the mathematical genius of their authors" may not far from the truth. The authors of sangaku are looking to gain notoriety through the publicity that the shrine or temple provides. But was the bemused Meijizen the target audience?

More on that in a couple of weeks. Below the cut is the solution to last week's problem.

The solution to the first problem (below the cut in this post) is the key. Name K, L and M the intersections of the three circles with the horizontal line. Then, by using that previous result,

Indeed, as in that problem, we can construct three right triangles, ABH, ACI and BCJ and apply Pythagoras's theorem in each.

Now, it suffices to note that KL = KM + LM, so

or, dividing by 2*squareroot(pqr), we get the desired result:

Inverting and squaring this yields the formula for r:

This gives us the means to construct this figure on paper using a compass and a marked ruler. Having chosen two radii p and q and constructed the two large circles (remember that AB=p+q) and a line tangent to both, placing M and C is done after calculating the lengths IK=CM=r and IC=KM=2*sqrt(pr).

Sangaku Saturday #13

Last week, we uncovered this configuration which is also a solution to our "three circles in a triangle" problem, just not the one we were hoping for.

This is something that happens in all isosceles triangles. Draw the inscribed circle, with centre B, and the circle with centre C, tangent to the extended base (ON) and the side [SN] at the same point as the first circle is. Then it can be proved that the circle with centre A, whose diameter completes the height [SO] as our problem demands, is tangent to the circle with centre C.

But that's not what I'm going to concentrate on. Despite this plot twist, we are actually very close to getting what we want. What the above configuration means is that, returning to the initial scaled situation with SO = h = 1 and ON = b, we get

Knowing a solution to a degree 3 equation is extremely powerful, as we can factor the polynomial and leave a degree 2 equation, which has simple formulas for solutions. So, to finish off, can you:

1: prove that

2: solve the equation 2x²-(s-b)x-1 = 0, and deduce the general formulas for p, q and r that fit the configuration we are aiming for;

3: test the formulas for an equilateral triangle, in which s = 2b.

This last question is the one the sangaku tablet claims to solve.

Grand Geroldseck castle - only a cat of different coat

Close to impressive Haut-Barr castle, a one-hour hike from Saverne, sit two more ruins. All of these castles were built around the same time, late 10th to early 11th century, but despite being so close, they weren't owned by the same people.

While Haut-Barr was under the control of the Bishop of Strasbourg, the two Geroldseck castles, the Petit and the Grand, were built by the Geroldseck family, in charge of protecting the lands of the Abbey of Marmoutier. At the time, Alsace was part of the Holy Roman Empire and divided into many largely independent pieces, so these castles facing each other were on a border of sorts. However, the male Geroldseck line went extinct at the end of the 12th century, and the land was co-owned by so many people that no-one was maintaining the castle. The last stand came in 1471, when a group of disgruntled knights used it as their base. The Imperial bailiff laid siege, won and the castle was left as a ruin after that.

While Haut-Barr castle gets a lot of visitors, owing to the possibility of driving there, the Grand Geroldseck is worth the extra walk and brief climb from its neighbour. As well as the dungeon, lots of walls and rooms are still present, making it an interesting place to explore. The remaining walls continue to receive restoration work - there seem to be a few differences between my first visit with @teamroquette and my second this summer, for example, I don't remember seeing the little garden a few years ago.

All that's left to say is: "OI YOU!... YES, YOU! Have a good time."

The best timepiece in the world (IMO): Strasbourg Cathedral's Astronomical Clock

I could go on about this thing for ages. There's so much history, so many symbols to spot, and so much information on display... This is going to be a long one.

I guess I'll start with the artistic aspect on which I have the least to say because it's the least up my alley. There's loads of mythology and Christian symbolism going on on this 18-metre tall monument, and these are the main draw for the general public, because they move around.

Like cuckoo clocks in neighbouring Schwarzwald, this astronomical clock has automatons. Every quarter hour, the lower level of the photo above sees a change of "age": a child, a young man, an adult and an old man take turns to be in the presence of Death, whose bells toll on the hour. At high noon, the upper level also moves, with the 12 disciples passing before Christ, and the rooster at the very top crows.

Moving on to what really makes me tick: the amount of information on this clock is incredible. The time, obviously, but actually two times are on display on the clock at the bottom of the picture above: solar time and official time. Given Strasbourg's position in the time zone, there is a 30-minute discrepancy between the two. Then there's all the astronomical stuff, like the phase of the Moon (just visible at the top), the position of the planets relative to the Sun (middle of the picture), a celestial globe at the base (pictures below, on the right)...

The main feature behind the celestial globe is another clock displaying solar time, with the position of the Sun and Moon (with phases) relative to the Earth, sunrise and sunset times, surrounded by a yearly calendar dial. These have remarkable features, such as the Moon hand that extends and retracts, making eclipses noticeable, and the calendar has a small dial that automatically turns to place the date of Easter at the start of each year. This sounds easy, but look up the definition of Easter and note that this clock is mechanical, no electronic calculating power involved! Either side of the base, the "Ecclesiastic Computer" and the "Solar and Lunar Equations" modules work the gears behind these features.

The accuracy of this clock and its ambition for durability are truly remarkable. Relative to modern atomic time, it would only need adjusting by 1 second every 160 years, and it correctly manages leap years (which is not as simple as "every 4 years"). It just needs winding up once a week.

Finally, the history. The monumental clock was built in the 16th century, and used the calendar dial above, now an exhibit in Strasbourg's city history museum. It slowly degraded until the mid-19th century, when Jean-Baptiste Schwilgué restored the base and upgraded the mechanisms. The "dartboard" on the old dial contained information like the date of Easter, whether it is a leap year, which day of the week the 1st January is... - all of which had to be calculated by hand before the dial was installed! - and was replaced by the Ecclesiastic Computer, which freed up the centre space for the big 24-hour clock, complete with Solar and Lunar Equations.

As you may have gathered, I am a massive fan of this clock. Of course, nowadays, all the imagery and information would easily fit into a smart watch, but a smart watch isn't 18 metres tall and powered by gravity and gears!

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.

Kaub

A few kilometres down the Rhine from Bacharach and on the other side, is the small town of Kaub. Its main attraction is the Pfalzgrafenstein, a remarkable, vaguely boat-shaped "castle" in the middle of the river. I use the term "castle" a bit loosely, as it wasn't a feudal lord's residence, rather a customs office on the river, with Gutenfels in the background providing a wider watch. A tollgate, basically. Sidenote, if you find other photos of this place, you'll notice how low the Rhine is right now...

While Burg Gutenfels in the hills has been converted into a hotel, not unlike Burg Stahleck discussed last time, the Pfalzgrafenstein has been preserved, and is open as a museum on some days of the week, with a small ferry allowing visitors to get there. Beside that, a car ferry runs all day between the two banks of the Rhine, so I could easily cross to quickly walk around Kaub.

Like a lot of riverside towns and villages in the area, Kaub is a charming place with many old buildings. It provides a picturesque backdrop to all the traffic going by, boats and trains, and the Rechte Rheinstrecke, the railway line between Wiesbaden and Koblenz on the right-hand side of the Rhine, was seeing more trains than usual, as the section between Mainz and Bingen on the Linke Rheinstrecke was closed. Long-distance trains, like this international EuroCity service bound for Zürich, were (and at time of writing probably still are) diverted via Kaub.