Busy Week Done! Lots Of Work Especially On Wednesday, And Helping Vent D'Est Organise Their Mah-jong

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...

Some life in the rock pools of Hashigui-iwa! The crabs in the first picture were very small, but the one hiding in the second picture was more sizeable. I forget how big, but it was big enough to observe scuttling for shelter as I approached. Closest match on iNaturalist appears to be the Striped Shore Crab, which grows up to 5 cm.

The local birds of prey, likely kites, were also out, surveying the area.

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.

Freiburg's Schwabentor

The Germany city of Freiburg im Breisgau, on the transition between the Rhine valley plains and the hills of the Black Forest, was part of the Duchy of Swabia until it dissolved in the 13th century due to the ducal line going extinct. It was around this time that its "Swabian Gate" was built at the Eastern edge of the town, facing the Swabian heartland.

Like Schaffhausen's Schwabentor, it has undergone upgrades and downgrades, taken damage and been restored over time. The current illustrations on the tower include St George slaying the dragon (1903) on the outside, and a merchant with a cart (first painted in 1572) on the inside, just visible in the picture below.

Freiburg's Altstadt has many gorgeous, colourful houses decorated with trompe-l'oeil facades. An effort has also been made to preserve the little rivers in the streets, known as Bächle. Local superstition says that anyone who accidentally steps in a Bächle will marry a local - unusual to see a place that values clumsiness!

Another short one today, just a couple of Christmas decorations from Strasbourg. The "tree of cathedrals" was, as far as I can remember, new for last year in front of the station, and is back again this year. I definitely should talk at length about the cathedral at some point... Not to worry, normal nerdy and rambling service will soon be resumed.

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.

July 1830: the Revolution I forgot

This is Bastille square in Paris. As anyone who's had history classes in France will know, this is Bastille as in Bastille day, 14 July 1789, when Parisians raided the Bastille prison to get weapons for their revolt against the king - the flashpoint of the French Revolution.

It's also rather well known in France that the Bastille prison was demolished shortly after, as Paris rid itself of symbols of the Old Regime. So it would make sense that this monument commemorates that, right? It's super famous, after all.

Wrong. This column commemorates the events of July 1830, some forty years later, the significance of which, I'll admit, I had forgotten.

So here's how it goes. Since 1789, France had oscillated between fragile compromises of constitutional monarchy, revolutionary fanaticism and the iron fist of Napoleon. Following the defeat of 1815, Paris entered a period of calm acceptance under King Louis XVIII, but his successor, Charles X, wanted to go back to the old ways.

So, in July 1830, Paris revolted again. Disposing of the king was a surprisingly quick affair, as in just three days, Charles X was gone. He was replaced by his cousin, Louis Philippe, who seemed more willing to placate the bourgeoisie. A new constitution was drawn up, known as the Monarchie de Juillet, or July Monarchy.

In this context, a monument to the victory of 1830 was commissioned, and this is it: the Colonne de Juillet (July Column), a 47 metre-tall column adorned with the names of the fallen revolutionaries, a mausoleum at the base and the Spirit of Freedom on the top - and is that camera surveilling the street below?

Louis Philippe had ascended to the throne after a revolution, but he would also descend from the throne after the next. In February 1848, Paris revolted for a third time, swiftly ending the July Monarchy and establishing the Second Republic... which, within just 4 years, would become the second Bonaparte dictatorship.

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.)

Sangaku Sunday Bonus - what about the other problems?

In the sangaku series, we've solved two of the four problems on this tablet, the middle two, which I believe were the easiest to work on in terms of geometric arguments - we hardly ever used more than Pythagoras's theorem, though the second one needed some more advanced algebra to finish off.

Here's a quick look at the problems at each end of the tablet, and the main ideas I had to solve them.

On the far left, we have two circles tangent to one another (with centres A and B), inside a larger circle (with centre O) so that their diameters add up to the diameter of the largest. The radii of these three circles, respectively p, q and p+q, are known. The unknown is the radius r of the circle with centre C, which must be tangent to all three original circles (it has a twin on the right-hand side with the same radius).

This is quite quick to solve. Remember that tangent circles mean that the distances between centres is equal to the sum of the radii, e.g. AC = p+r, BC = q+r... Al-Kashi's theorem, which is a general version of Pythagoras's theorem, links the lengths of three sides of a triangle with one of the triangle's angles, and the triangles CAO and CAB have an angle in common, which yields the equation for r by isolating this angle in each application of Al-Kashi's theorem. The result is:

The problem on the far right seems to start in a similar fashion: two circles with fixed radii are offset by a fixed distance. A third circle has its diameter equal to the remainder of the diameter of one of the large circles: this radius can be calculated with little difficulty. What we want to do next is construct circles which are tangent to the two large ones, and the one previously constructed.

The radius of the circle with centre C1 can be obtained as above, but this method does not seem to extend to the subsequent circles, as O, D and C1 are no longer aligned, and there no longer appears to be a common angle in the triangles we want to work with. So I went for a parametric approach, understanding the curves that contain points that are equidistant from two circles. The red curve (which looks like a circle but isn't one) is the set of points at equal distance from the two largest circles, and we seek to intersect this with the set of points that are at equal distance from one large circle and the smaller one, the green curve. The intersection is equidistant from all three circles, so it is the centre of the circle we want to construct. Rotate and repeat for subsequent circles.

The general formulas are horrible and not worth showing, but this is another problem where I have been able to read the results on the tablet. The large circles have radii 61 and 72, and the offset is 23. The radii of the smaller circles, starting with the one in the middle and working outwards are:

17, 15.55, 12.292, 8.832 and 6.038 (I see 八, but I'll give the authors the benefit of the doubt as the top of the character 六 may have been erased by time)

The results with our exact formulas are:

17, 15.58, 12.795, 9.076 and 6.444

Rather close! As with the "three circles in a triangle", I do not know how the authors originally solved this problem.

Two crossings of the Rhine

The Swiss city of Basel lies on the border with France and Germany, and, as it's Switzerland, it hasn't changed hands or been attacked much (though the French did use Basel as target practice for a new cannon from their fort at Huningue once). It has a well-preserved historic centre, and, with the Rhine's current being consistently strong, it has a rare form of transportation.

This little ferry has no motor. It is tethered to a wire that crosses the river, and a lever at one end of the tether on the boat is all that's needed to turn the boat into the current which does the rest.

It's incredibly simple and easy! For a more engaging version of the story, here's a video by The Tim Traveller.

While I rode the boat with my sister, I continued upstream alone to another crossing, a bridge which doubles up as a dam for hydro-electric power stations on either side - or Kraftwerk as it's known in German.

Unlike in Basel, the Rhine at this point is an international border: Germany on the right-hand side, and Switzerland on the left-hand side. But with Germany and Switzerland being signatories of the Schengen agreement, this is what the border looks like:

The Rhine sees some impressive barges navigate roughly between Schaffhausen and Rotterdam, so there is a rather impressive lock next to this dam and the Kraftwerken. This is the view downstream from the top of the lock, with what I suspect was a border post on the right? I don't know, but I seem to remember that black and white stripes had some significance.