Before Hello Kitty, It Was Evangelion

I mentioned a train accident that was local to the Strasbourg area - here's an air crash on Mont Sainte Odile that truly shocked the Alsace region. There is a memorial on the mountain, I should visit and pay respects some day.

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.

Sangaku Saturday #14 - the grand finale!

We are only a few steps of algebra away from solving the "three circles in a triangle" problem we set in episode 7. This method will also yield general formulas for the solutions (first with height 1 and base b; for any height h and half-base k, set b=k/h and multiply the results by h).

Before we do that, it's worth noting what the sangaku tablet says. Now I don't read classical Japanese (the tablet dates back to 1854 according to wasan.jp), but I can read numbers, and fishing for these in the text at least allows me to understand the result. The authors of the sangaku consider an equilateral triangle whose sides measure 60: boxed text on the right: 三角面六尺, sankaku-men roku shaku (probably rosshaku), in which 尺, shaku, is the ten marker. In their writing of numbers, each level has its own marker: 尺 shaku for ten, 寸 sun for units, 分 fun for tenths and 厘 rin for hundredths (毛 mô for thousandths also appear, which I will ignore for brevity). Their results are as follows:

甲径三尺八寸八分六厘: diameter of the top (甲 kou) circle 38.86

乙径一尺六寸四分二厘: diameter of the side (乙 otsu) circle 16.42

反径一尺二寸四分二厘: diameter of the bottom (反 han) circle 12.42

I repeat that I don't know classical Japanese (or much modern Japanese for that matter), so my readings may be off, not to mention that these are the only parts of the tablet that I understand, but the results seem clear enough. Let's see how they hold up to our final proof.

1: to prove the equality

simply expand the expression on the right, taking into account that

(s+b)(s-b) = s²-b² = 1+b²-b² = 1.

2: the equation 2x²-(s-b)x-1 = 0 can be solved via the discriminant

As this is positive (which isn't obvious as s>b, but it can be proved), the solutions of the equation are

x+ is clearly positive, while it can be proved the x- is negative. Given that x is defined as the square root of 2p in the set-up of the equation, x- is discarded. This yields the formulas for the solution of the geometry problem we've been looking for:

3: in the equilateral triangle, s=2b. Moreover, the height is fixed at 1, so b can be determined exactly: by Pythagoras's theorem in SON,

Replacing b with this value in the formulas for p, q and r, we get

Now we can compare our results with the tablet, all we need to do is multiply these by the height of the equilateral triangle whose sides measure 60. The height is obtained with the same Pythagoras's theorem as above, this time knowing SN = 60 and ON = 30, and we get h = SO = 30*sqrt(3). Bearing in mind that p, q and r are radii, while the tablet gives the diameters, here are our results:

diameter of the top circle: 2hp = 45*sqrt(3)/2 = 38.97 approx.

diameter of the side circle: 2hr = 10*sqrt(3) = 17.32 approx.

diameter of the bottom circle: 2hq = 15*sqrt(3)/2 = 12.99 approx.

We notice that the sangaku is off by up to nearly a whole unit. Whether they used the same geometric reasoning as us isn't clear (I can't read the rest of the tablet and I don't know if the method is even described), but if they did, the difference could be explained by some approximations they may have used, such as the square root of 3. Bear in mind they didn't have calculators in Edo period Japan.

With that, thank you very much for following the Sangaku Weekends series, hoping that you found at least some of it interesting.

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.

Summer highlight reel: Oberkirch

I praised this hike on a loop South of Oberkirch for its amenities, but the views were also fantastic. In chronological order, here's the first vineyard I came across on the circuit.

The first drinks hut, with schnaps, is the Köbelesberghof to the left, out of frame. The hamlet opposite, which OpenStreetMap names In der Rot, looks gorgeous among the vineyards and forest!

Just below the summit area of the Geigerskopf is the Busseck Hof vineyard, and beyond, the plains in which the Rhine flows.

Turn around at the previous photo and the path to the Geigerskopf summit will appear. A tower at the top offers some stunning, unimpeded panoramas of the Rhine plains and the Vosges on one side (Strasbourg is visible in this picture), and the Black Forest hills on the other.

Finally, climbing down, past Busseck and past the drinks hut I stopped at (Klingelberger Hütte), we reach this viewpoint overseeing the town, with the castle visible on the hills opposite. It seems that all the fruits are grown here: apples, plums, pears and grapes...

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!

February 2020 was weird

In my mind, visiting Oberkirch mid-February 2020 as my last outing before lockdown in Europe. Actually, that's not true - late February I went to Western France and Paris... but this was indeed my last solo excursion. I went for the carnival, which did take place, but Covid was very much on the horizon at that point. So there's something special about looking back at seeing people at the castle, enjoying the views and nature while it was still possible.

I sighted this Vectron at Karlsruhe: a Dual Mode, which can move either with its Diesel engine or getting its energy from the overhead wires (German 15 kV AC only for now). However, it only delivers a third of the power of a standard all-electric Vectron, and is therefore not designed for main line hauling, and is expected to be more at home near sorting yards. Deutsche Bahn have also bought some of these as ICE rescue locomotives, serving when a train breaks down.

For Vectron!

Produced since 2010 by Siemens, the Vectron is a modular locomotive platform with various engine options - AC electric, quad-voltage for use across Europe, "last-mile Diesel" option for parking, Diesel motors, dual mode/hybrid... It hauls both freight and passenger trains. But the main reason I've wanted to mention the Vectron is...

this Mitchell and Webb sketch!

This is from series 3 of That Mitchell and Webb Look, which was aired in 2009. The Siemens Vectron was officially launched in 2010, so it's fair to say that the name appearing in both is a coincidence. However, when I see a Vectron, it reminds me of this sketch, so it's harder for me to take this train seriously!

But it is serious business, as it is one of the most common locos in continental Europe. Only Iberia (due to using a different gauge) and France (because if it ain't Alstom, they'll oust 'em) don't see much of them. The examples shown here are from Germany, Switzerland and Slovakia, and were all pictured in the same area of Germany. The quad-voltage version in particular allows companies to carry freight all over Europe, they're virtually borderless.

Yet here I am, still snickering at the name, by Vectron's beard!

Japan's biggest shimenawa: Izumo Taisha Kagura-den

We've seen larger and larger sacred ropes in our previous posts, and here's the largest of all: the shimenawa adorning the Kagura-den at Izumo Taisha. It is 13.5 m long and weighs 5 tons!

Shimenawa ropes are made with hemp or rice straw. This shot shows just how densely packed Izumo Taisha's shimenawa is - it's almost like a tree trunk, truly an impressive and imposing feature.

Here and in previous examples, we can see folded paper shide streamers, another symbol of sacred items in Shinto, on the first picture. Knots also appear, particularly on the ropes on the Meoto Iwa at Futami, as paper would not fare well by the sea... These ropes and streamers indicate a demarcation line between our world and the realm of kami spirits.

The Kagura-den is a hall just outside the main compound of Izumo Taisha, to the West. It was built to house kagura rituals, traditional dances which retell the stories of the early gods. As Izumo, formerly in Iwami province, is one of the most ancient shrines in Japan, Iwami Kagura is one of the major forms of the dance.

As a bonus, here's a train decorated with characters of Iwami Kagura.

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.