World's Loudest Prius

Toba Castle ruins

After Hikone Sawayama and Numazu Nagahama, a final entry in my mini-series of castles that are outside the Top 100 and Next 100 lists - until I get to see more! - is Toba Castle, located on the glorious Shima coast, and of which little beyond a few walls and foundations are left.

Now a park, the top level offers some good views of the coastline, only a short climb up some stairs from Toba's attractions sea level. Admittedly, the best views require a longer hike, and when I visited, Shiroyama Park was at the end of quite a long day's walk!

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!

Temples on the high street: Kyôto Teramachi and Shinkyôgoku

A typical Japanese covered high street, right? Yes, there are lots of shops left and right, but this is Kyôto, a millennial capital and centre of cultural and religious tradition in Japan. So what happens when a modern high street encounters a centuries-old temple, bearing in mind that it would be very bad form to ask the priests to sell up?

Well you leave the temple alone and build around it. So, in between the big name drugstores, Animate, clothes shops and cafés lined with the street's signature bricks and tile floor, here's the entrance to Seishin-in, with traditional wooden doors and tile roofing. It is also just visible in the first picture - see if you can notice it.

Seigan-ji is easier to spot, further illustrating the contrast. Online maps show that this temple even has a cemetery in the middle of the next block, completely encircled by shopping streets and businesses.

It's a similar situation for Tenshô-ji, though, this far up the high street, the commercial tissue becomes less dense. These temples seem a bit larger and own more land. Still, we've gone from a towering, mineral, covered street in the top photo, to an open path with low buildings and plenty of vegetation, with no transition.

These scenes of coexistence fascinated me when I first visited Kyôto in 2016, with a hotel in this area, so it was great to see them again on my brief return to the city in 2023. More fascinating still is the fact that one of these "just off the high street" temples is not only very old, but hugely historically significant... and I'm yet to mention it!

A ceux qui pensent "on n'a jamais essayé": si si, on a essayé. Et ceux qui en sortent nous laissent assez invariablement les mêmes messages.

Out of service: the Trieste-Opicina tramway

While the reopening of Notre-Dame cathedral in Paris is making big news, and while I'm in a bit of a tram phase on the blogs, spare a thought for the tram line between Trieste and Opicina, closed following an accident in 2016. And it's a real pity, because it was a wacky one.

Trieste is a city by the Adriatic Sea, surrounded by steep hills - and I mean steep. Opicina is 300 m higher, and the tram line features gradients as steep as 26% - link to the Hohentwiel hike for scale. Steel wheels on rails weren't going to be enough...

Initially, the steepest section was built as a rack-and-pinion railway, but in the late 1920s, it was replaced by a funicular system. Cable tractors would be coupled to the streetcars to push them up the hill, and control their descent on the way down - that's the curious boat-like vehicle in the photos (at least I'm getting boat vibes from it). The picture below shows just how steep the climb is.

In the later years of operation, these cable tractors were remotely controlled from the tram. The streetcars themselves date back to 1935, with wooden doors and fittings, making the Trieste-Opicina tramway a charming and technically unique heritage system.

Sadly, the line is not running. Two streetcars collided in 2016, they were repaired, but service has not resumed. One vehicle, coupled to the cable tractor, remains stationary at the foot of the climb, near where the second photo was taken. A look on Google Street View shows that cars are now habitually parked on the disused tracks. The number 2 tram route between Trieste and Opicina is currently served by the number 2/ bus.

Sangaku Saturday #9

As in every odd-numbered episode, we're going to set a problem - the next stage towards solving the "three circles and a triangle" sangaku. We are looking for one more equation between the radii p, q and r, it will be obtained with a similar method to the previous step... but the formulas will be a bit longer, so roll your sleeves up and don't be scared!

Here are the lengths we know:

SO = 1 , SN = b , SA = p , BO = q , CQ = r and

Here is also a list of known pairs of perpendicular lines:

(SO) and (ON) , (SO) and (PC) , (ON) and (CQ) , (SN) and (CR).

[P, Q and R are defined as the orthogonal projections of C onto the sides of SON.]

The equation we are looking for will come from getting two expressions for the square of the length CN.

You can work out how to do this by yourself if you feel like it, or check below the cut for the steps and to check your result. As always, details and a bit of history next week!

1: After working out the length QN, get a first expression of CN² by using Pythagoras's theorem in the right triangle CQN.

2: Proceed similarly in the cascade of right triangles CPS, CRS and CRN, to get a second expression of CN².

Conclude that

Up the Monumental Stairs at Auch

Some 75 km West of Toulouse, the city of Auch is far away enough for red brick to be far less prevalent in buildings. It developed along the Gers river, with the higher-ups living... well, higher up.

The Monumental Stairs were built in the 1850s when, following a rebellion against Napoleon III's coup installing the Second French Empire, the prefect decided to give the townsfolk something to do (per the city council, "créer des chantiers afin de donner de l'ouvrage à ceux qui en manquent"), rather than just repress. The results were a water and gas distribution network, and the Monumental Stairs, creating a comfortable link between the upper town and the riverside 35 metres below. Later, a statue of d'Artagnan, a musketeer made famous by Alexandre Dumas novels, was added.

Behind d'Artagnan here, rises the Tour d'Armagnac, a 14th-century prison. Unfortunately, it is privately owned and cannot be visited, unlike the neighbouring cathedral, built between 1489 and 1680.

While the back of the cathedral, visible in the top photo, is clearly gothic, which fits the start of construction, the front facade is in a later, classical style. This would fit the timeline, as cathedral building usually started with the crypt and the altar, working outwards, and finishing with the massive entrance and towers. Walking away, further West, we encounter one more figure of the town: Intendant Mégret d'Étigny, who administrated the Auch-Pau area under King Louis XV, and is credited with infrastructure improvements in the region at the time.

French and German cross-border railcars

Just on the French side of the border, Wissembourg station sees French TERs arriving from Haguenau and Strasbourg, meeting German Regionalbahn from Landau and Neustadt. When a French service doesn't cross the border itself, SNCF and DB services are often synced up, giving us the chance to see both companies' cross-border efforts side by side.

The B 85500 is a brand-new bi-mode (electric and Diesel) multiple unit from the Alstom Régiolis family. While not the first international Régiolis - a tri-voltage electric version runs between Évian and Geneva -, the B 85500 adds autonomous Diesel power. With 30 units on order, it aims to revive and/or intensify cross-border routes into Germany. I was under the impression there was a bit of a gathering at the front end of the train, maybe local politicians marking the type's first visit to Wissembourg?

On the German side meanwhile, we have a standard railcar for what DB regard as a relatively low-density non-electrified regional route, which was closed to passengers for over 20 years between 1975 and 1997. The BR 628/928 (628: power car, 928: trailer car) is a 1980s design. Besides the lack of low floor, there's not a lot wrong with them, and they have comfortable, current DB Regio interiors. More recent types have been used, and if a plan to electrify Landau and Winden stations to allow charging of battery-EMUs goes ahead, there could soon be the latest trains on the German side of Wissembourg station too.

Which brings us to our "I beg your pardon" of the week:

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 #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.