Heading Towards Tokyo, Two High-speed Shinkansen Lines Join Up Roughly 8 Km North Of Ômiya. I Am Riding

Sangaku Saturday #8

Having established that sangaku were, in part, a form of advertisement for the local mathematicians, we can look at the target demographic. Who were the mathematicians of the Edo period? What did they work on and how?

The obvious answer is that the people in the Edo period who used mathematics were the ones who needed mathematics. As far back as the time when the capital was in Kashihara, in the early 8th century, evidence of mathematical references has been uncovered (link to a Mainichi Shinbun article, with thanks to @todayintokyo for the hat tip). All kinds of government jobs - accounting, such as determining taxes, customs, or engineering... - needed some form of mathematics. Examples above: 8th-century luggage labels and coins at the Heijô-kyô Museum in Nara, and an Edo-period ruler used for surveying shown at Matsue's local history museum.

As such, reference books for practical mathematics have existed for a long time, and continued to be published to pass on knowledge to the next generation. But sangaku are different: they are problems, not handbooks.

More on that soon. Below the cut is the solution to our latest puzzle.

Recall that SON is a right triangle with SO = 1 and ON = b. These are set values, and our unknowns are the radii p, q and r of the circles with centres A, B and C. While these are unknown, we assume that this configuration is possible to get equations, which we can then solve.

1: The two circles with centres B and C are tangent to a same line, so we can just re-use the very first result from this series, so

2: Also recalling what we said in that first problem about tangent circles, we know that

Moreover, PA = AO - OP = AO - CQ = (p+2*q) - r. Thus, using Pythagoras's theorem in the right triangle APC, we get a new expression for PC:

since 2(p+q)=1 (the first relation). Equating the two expressions we now have of PC², we solve the equation for r:

again using the first relation to write 2q-1 = -2p.

It only remains to find a third equation for p to solve the problem.

Pouancé

The small town of Pouancé is on a remarkable geographical "quadruple point", on the border of four départements! A peculiarity that dates back to the Middle Ages, when it was on the battlefront between France-affiliated Anjou and the still fiercely independent Duchy of Brittany. As such, Pouancé had a castle; its Breton counterpart was at Châteaubriant, and both towns were besieged at some point during the 15th century.

With a friend from Châteaubriant, we got to witness barriers being removed as what we guessed were maintenance or renovation works on the path around the castle were finished. The castle itself only opens during the summer, but at least we got to walk all around and get some good views of the castle, through the neighbouring park.

If you're driving into Pouancé from the West, this is how you know you've arrived:

Hormone is apparently short for ホルモン焼き, horumon-yaki, a dish that originated in Ôsaka. It is made from miscellaneous organs, but the organs aren't where the hormones come from... It is marketed as a meal that can improve stamina, but ホルモン is also close to 放る物, read hôrumon in the local dialect ôsaka-ben, which means "discarded things", which is what horumon-yaki is made of. Maybe the latter came first, and was construed in to ホルモン as a kind of joke.

With thanks to @felvass for the hint.

Friendly faces on Dôtonbori

Dôtonbori is the street to go restaurant crawling in Ôsaka (if you have the stomach). As there's a lot of venues, there's a lot of competition, so a lot of wacky stuff to draw the passer-by's attention.

If anyone knows why this restaurant is called Shôwa Hormone, please let me know. Shôwa, I can guess, is nostalgia for the post-war Shôwa era; but Hormone needs a good story behind it!

Is this guy mad at people double-dipping their fried skewers?

By the way, that's two fronts featuring another monument of Ôsaka, Tsutenkaku tower, just in case you forgot where you were.

Finally, we have this guy, a true local hero: Kuidaore Tarô. This animatronic was introduced in 1950 as a mascot for the Cuidaore restaurant, which has since closed, but Tarô and his drumming were such a stable of Dôtonbori, that people clamoured to have him back.

I dunno. I think he looks like Brains from Thunderbirds under the influence of the Mysterons. A figure of his time though.

"Kuidaore" by the way, is from the proverb:

京都の着倒れ、大阪の食い倒れ Kyôto no ki-daore, Ôsaka no kui-daore Spend all your money on clothes in Kyôto, and on food in Ôsaka

Today, "kuidaore" is colloquially translated as "eat until you drop" - so go restaurant crawling if you can!

Three castles above Éguisheim

The village of Husseren-les-Châteaux is a peculiar one: at 1.2 km², it is the smallest commune in Southern Alsace, totally surrounded by Éguisheim. Beyond the vineyards, in the hills above the village and on the border with Éguisheim, are three castles, separated by... nothing.

Unlike other places where several castles can be found, such as Andlau or Ribeauvillé, this was only one unit, with the Dagsbourg and Weckmund being extensions of the original Wahlenbourg in the middle. Each section had its own dungeon.

I visited with my sister on a very overcast day two years ago, with low cloud descending on the hills. It made for some moody shots. We weren't alone up there - which was probably helpful! Also we used a car, and the car park isn't far, so it wasn't a creepy hike up or down.

As for the castle itself, it was destroyed during the Six Deniers War in 1466. The Habsburgs intended to conquer Mulhouse, and used the flimsy pretext of a miller being owed six deniers to start the invasion. But Mulhouse found allies in neighbouring Swiss cantons (before the Swiss Confederation was a thing) and won, taking out the fortress of Éguisheim along the way... as well as the miller whose complaint gave the Habsburgs the excuse they were waiting for.

"Stick an aircraft engine in it" part 1 - Boeing Jetfoil

In the late 60s and early 70s, all branches of transport were hoping for an increase in performance similar to what the jet airliner brought to aviation, and the solution was invariably to use similar gas turbine technology, with invariably identical career trajectories when the oil crises hit, as, apart from in aviation, far more economical engine options were available. So I was very surprised to see this still active in Japan last summer:

This is a hydrofoil which uses gas turbines to power a pump-jet. Once it is going fast enough, it takes off and runs on foils, greatly reducing water resistance and achieving speeds up to 45 knots, over 80 km/h (which, on water, is very fast). I remember seeing exactly this type of vessel in ferry brochures when I was a child; Oostende Lines operated some between England and Belgium. The advent of the SeaCat, a class of huge Diesel-powered car-carrying catamarans, got the better of the hydrofoils and the hovercraft, which was incidentally another case of "stick an aircraft engine in it".

This specific class of hydrofoil takes the mantra to another level, as it was designed by Boeing, which named it the 929 Jetfoil. Production was licensed to Kawasaki Heavy Industries in Japan, which made boats for the domestic market. The Rainbow Jet is one of these, running between Sakaiminato on the San'in coast and the Oki Islands. I saw more of them at Atami in Eastern Shizuoka, providing transport to the Izu Islands. So, despite the astronomical 2150 L/h consumption (though to be fair, I can't find consumption numbers for equivalent foot passenger-only catamarans), Japan still runs them...

The "Witch's Eye" - Engelbourg Castle ruins, Thann

For what it's worth, Prince Albert II of Monaco is visiting Thann in the South of Alsace this weekend, as he is also Count of Ferrette. Held by the Austrian Habsburgs for a while, the title was bestowed upon Cardinal Mazarin, a prime minister to King Louis XIV, following the French conquest of Alsace in the 1640s. A member of House Mazarin married the Prince of Monaco in 1777 and that's how the title ended up where it is today. I don't plan to explain the origin of the other bazillion oddball titles the Prince of Monaco holds...

It's just a pretext to mention Engelbourg Castle, built by the Counts of Ferrette in the 13th century on a hill above Thann. The French Counts wouldn't benefit from it for too long: as it lay too far from the German border to be of strategic value, the same Louis XIV ordered its demolition in 1673. But, as you can probably tell from the photo, something weird happened during the process. The cylindrical dungeon came apart in segments, and one of these landed on its side. For whatever reason, it was left there, creating a unique landmark known as the "Witch's Eye", as an echo to the "Witch's Tower" in the town of Thann, but all it reminds me of is Polo mints.

A short hike uphill from Thann, it also offers nice views of the valley and the vineyards on the neighbouring hills.

Back to the Arzviller-Henridorff lock ladder

I took my own advice and returned to Arzviller to walk the full section of closed canal. It turns out the upper part is less overgrown and more active than the lower end. I'd only gone from the bottom lock, n°17, up to lock n°14 (under the railway bridge) at the time.

Starting from the top, the house at lock n°2 is a crêperie restaurant, while lock n°4 is a gîte complete with a private fishing pier. I counted four lock houses with some form of art going (n°2, 10, 13 and 15). The other lock houses are more or less well-maintained residences. There are no houses standing at locks n°14 and 17.

The Vallée des Éclusiers, or Lock-keepers' Valley, is well worth the wander. The quickest is either to drive there or cycle from Lutzelbourg station. But more on the logistics tomorrow - for now, let's enjoy lock n°9, probably the most dramatic on the route.

History on the high street: Honnô-ji

Similarly to the other examples mentioned in a previous post, a temple sits at the North-East end of Teramachi shopping street, with traditional-style gate and buildings. But this is more than an ordinary temple, as it bears the name of a pivotal moment of Japanese history.

I say "bears the name", because this is not the actual location of the Honnô-ji incident. The original Honnô-ji was built a few kilometres to the South-West, nearer to Shijô (the 4th East-West street South of the castle; the current temple is North of Sanjô, or 3rd street). It was there that, in 1582, Akechi Mitsuhide, a general in the force aiming to unify feudal Japan, rebelled against his leader, Oda Nobunaga. The importance of Oda Nobunaga would be its own series if I had the material, but if you know, you know; I'll concentrate on Honnô-ji.

Akechi's forces laid siege and burned the temple down, and it is widely believed that the trapped Nobunaga committed seppuku - however, his body was never found. Akechi, meanwhile, had underestimated other generals' loyalty to Nobunaga. Toyotomi Hideyoshi swiftly brokered a peace treaty with the lord he was fighting in the West to rush back to Kyôto, and ran Akechi down just two weeks after the coup.

Hideyoshi was accepted as the new leader of central Japan, and went back to expanding westward. He also ordered that Honnô-ji be moved to its current location and rebuilt in 1591. It has, in fact, needed to be rebuilt several times since, each time after fires ravaged the capital. Beyond its function as a Buddhist worship site, the new Honnô-ji includes a mausoleum to Oda Nobunaga, as the decision was made by his successors to honour him there.

I remember passing by this temple in 2016, during my first trip to Japan, a three-week work visit in Kyôto. I had no notions on Japanese history at the time, and it was only a while later that I realised the importance of Honnô-ji. So, as an aside to Gion Matsuri in 2023, I decided I had to go back for a closer look.

The fortifications of Wissembourg

From one fortified town to another, just South of the current French-German border: Wissembourg. In a region rife with conflict, between cities and lords, sometimes between a city and their own lord, protective walls, moats and towers around the town were a must.

This tower, called the Poudrière, was built in the 13th century, and served as gunpowder storage at some point, hence the name. The walls in front of it are more recent, dating back to the 16th century, featuring a dam system which would flood the moat if needed. This complex can be seen in the North-East corner of the town on this 1750 map.

Despite these protections, Wissembourg suffered massively between the 15th and 17th centuries. And in spite of all that, the town centre retains much of its original plan and many traditional buildings. I should go back on a nicer day to get better pictures...

Sangaku Saturday #3

Another problem this week, adding to the configuration we looked at previously.

Specifically, given two circles tangent to each other and tangent to a same line - these circles have respective centres A and B, and respective radii p and q -, we want to construct the circle tangent to both of the original circles, and tangent to the line beneath them.

Can you prove that the radius of this third circle, denoted r, satisfies

and deduce a formula for r as a function of p and q?

Help below the cut, answers next week.

Hint. Name K, L and M the intersections of the circles with the line below, and use the previous result on each pair of circles to get the lengths KL, KM and LM. One of these lengths is the sum of the two others.