Pouancé

Local Ampelmännchen!

While the East Berlin Ampelmännchen has become a cultural icon, other cities in Germany have been getting their own personalised red and green men for pedestrian traffic lights, from historical figures like Karl Marx in Trier, to mascots.

So on the left, we have Brezelbu, a mascot for Speyer's annual pretzel festival. These lights were implemented in time for the 2023 edition.

On the right, we have Det, one of six little characters introduced in the 1960s on TV channel ZDF to transition in and out of ad breaks. The characters were called the Mainzelmännchen, or the little men from Mainz, so the city has adopted them as mascots of sorts.

Éguisheim's super-slim houses

Just a few kilometres to the South-West of Colmar is the village of Éguisheim, with a preserved Medieval centre, featuring gorgeous timber-framed houses. Some of these are the daintiest, slimmest houses you can find!

It begs the questions "why?" and "how did people live in them?" Well, in some places with similar thin houses, taxes have been the reason, as residences were taxed based on how far they extended on the street, with few restrictions on height and depth. Heck, here's an example in Colmar that doesn't have any footprint on the street: the Muckekaschtele, or "fly box house". It has a surface of 25 m², but it wasn't originally used as a home - it was a watchtower to make sure people were paying customs when bringing goods to market.

I'm not sure if that's what's at play in Éguisheim, it just looks like they're making the most of the space available between streets. In any case, they're charming and eanred Éguisheim the title of "France's Favourite Village", a poll organised by public TV channel France 2, in 2013. Éguisheim's more historic claim to fame is having been the birth place of Pope Leo IX (reigned 1049-1054).

Sangaku Saturday/Sunday is taking a week off.

Running around Koblenz station the other evening, I couldn't resist the cute message on this local train. Instead of a deadpan "depot" or "not taking passengers", this train is "going to bed"!

ICE 3 (Gen.1)

After visiting Karlsruhe Palace, I had a bit of time while waiting for a (packed) regional train South to wander around the station. There was quite a lot going on, as on top of the local traffic from S-Bahns to REs, busy intercity lines to Freiburg and Basel, Stuttgart and Munich, Mannheim and Frankfurt meet here. A chance to see my favourite German high-speed train: the Baureihe 403 ICE 3.

Entering service in 2000, over 10 years after the first InterCity Expresses, the ICE 3 was a revolution in European high-speed rail. These were the first 300 km/h-capable trains that weren't of a "power car & carriages" layout, using distributed traction (motors all along the unit) like Shinkansen trains. And a sleek shape to boot! They were designed by Alexander Neumeister, who also penned the 500 Series Shinkansen. Hmm, maybe that's why.

The ICE 3 would be the first example of Siemens's Velaro series, which would be an export hit: the Netherlands, Spain, China and Russia purchased this model. The type received a redesign in the late 2000s that I could only describe as "more beefy": the Velaro D was taken up by Germany, Turkey and Eurostar.

The destination on that sweltering day I saw the Jetfoil was Jizôzaki and Mihonoseki lighthouse. It's just a 2 km walk from the village, but gosh it was difficult that day! Still, the views of the coastline were, as always, well worth the effort.

The lighthouse itself is a nice little building, built at the end of the 19th century, and nicely kept (if you can spot the ladies raking in front of the entrance in the picture). It's circled by a short walking trail.

The tower doesn't need to be too high as it stands on a cliff, while the former residence now houses a shop and a tea room with a view out to sea.

Like Shionomisaki I covered some time ago, Mihonoseki lighthouse has a "Guardian of Light" character designed for it! This one looks somewhat more confident than the pensive guardian of the South!

Japan's rail speed record holder: 300X

Sticking with the Tôkaidô Shinkansen and the SCMaglev & Railway Park, this is the (highly abridged) story of the fastest conventional train in Japan.

Following privatisation and sectorisation in the 1980s, and seeing France and Germany take the lead in the high-speed train department, the three JR companies that had Shinkansen lines set about catching up to offer 300 km/h services where they could. The aerodynamics and sheer weight of the venerable 0 Series and its derivatives weren't going to cut it, so each company designed a prototype train to test new technologies.

JR Tôkai's solution was 300X, officially Shinkansen Class 955 - numbers starting with 9 are trains not open to the public, either prototypes or work trains like Class 923 "Doctor Yellow". Launched two years after JR West's WIN350 and JR East's STAR21, it featured two radically different end cars. The more elegant one, in my opinion, is on display at JR Tôkai's museum in Nagoya, while the other is preserved at JR Group's research centre in Maibara. The intermediate cars have all been scrapped.

The three prototypes took turns to hold the national rail speed record, and, 300X being the last, it took the record last, and holds it to this day. We mentioned the fact that the Tôkaidô Shinkansen still had too many relatively tight turns, but the Maibara to Kyôto stretch is the best part, and that's where this train hit 443 km/h in 1996. This video may, or may not, be that run, but it still looks very fast - note the unusually large, "flying saucer" pantograph cowlings.

Unless JR East decide to go completely bonkers with their ALFA-X prototype, it's unlikely that the record is going to be beaten any time soon. It's not in the spirit of these trains, they are pure test beds and run quite extensively with the aim of increasing service speeds. Records also require special preparation of the tracks, which is why the French TGVs made their 1990 and 2007 record runs before the opening of a brand new line.

But JR Tôkai have gone much faster with their Maglev programme, which holds the world speed record for passenger trains outright with 603 km/h. Behind 300X at the museum is a predecessor of that record holder, MLX01, the first Maglev train to clock over 500 km/h. Again, this is not (just) showboating, the lengthy test programme's main aim is to prove that consistent service at very high speed with this technology is feasible, so that the Maglev Chûô Shinkansen can achieve this when it opens (if Yamanashi-ken can agree on a route).

Expo 70 Japanese Garden

While most of the pavilions of the 1970 World Expo at Osaka were demolished, a very large exhibit was preserved: the Japanese garden. Located in the North of the enormous Commemoration Park, it's designed as a stroll through the ages, with four sections inspired by different periods, Ancient, Medieval, Modern (Edo period) and Contemporary - kind of like the zones from The Crystal Maze, just without the puzzles and Richard O'Brien.

From rock gardens to ponds and rolling hills, from a bamboo grove to wisteria and maple trees, it hits the notes you'd expect from a summary of Japanese landscape styling, but allows one plenty of room to contemplate each tableau - I'm emphasised how big the overall park is, but the Japanese garden alone covers 26 hectares.

One of the few gripes I have with Koishikawa Kôrakuen, my favourite garden in Tokyo, is that some buildings are close and large enough to appear in the background. With the Expo pavilions gone however, most views in this park don't have that problem. Only the Tower of the Sun and the Osaka Wheel are able to rise above the park in this view towards the South.

As the maths problems take a break, maybe we can have a brief pub quiz. So...

Q. What's that building on the 10-yen coin?

A. The Phoenix Hall at Byôdô-in, Uji.

Initially built as a villa by a member of the Minamoto clan just before the year 1000, the land was sold not long after to members of a rival clan, the Fujiwaras, who turned it into a Buddhist temple named Byôdô-in in 1052. The most striking feature of the temple is the Amida Hall, which with time gained the name Phoenix Hall due to its overall appearance: the two outer corridors are the wings, and a corridor extending behind is the tail.

At the same time, tea production was picking up in Uji, and by the 14th century, Uji tea had become well renowned. I need to go back there someday, my first visit was just an afternoon flick after completing the climb of Mt Inari in the morning. I thought of going back there in the summer of 2023, but couldn't quite make time for it.

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.