Latest Posts by merpmonde - Page 3

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.

Space Port Kii (from a distance)

Japanese company Space One has been in the news recently for their second attempt at launching their rocket, Kairos - for Kii-based Advanced & Instant ROcket System; as far as acronyms go, I'd give it a 5/10, it's rather long-winded but has some good ideas at the right moments. The rocket, designed to be a cheaper option for lighter satellites, unfortunately didn't make it into orbit, losing control after 95 seconds.

The launch site is located on the North-East edge of Kushimoto, Honshû's southernmost city, its entrance building visible from the railway line. The action area is further into the woods, by the coast. I didn't visit the site obviously, but the entrance and some support posters in Kushimoto town were hints of the project's presence. They have a neat little mascot too, a space puppy!

Notre-Dame... de Strasbourg

I know, I know. Notre-Dame in Paris just reopened. But Notre-Dame is a very common name for churches in France. In fact, we covered one in Le Havre not that long ago, possibly one of the smallest cathedrals in the country. At the other end of the scale, one of the largest, if not still the largest, is Notre-Dame de Strasbourg. Built during the same time period as its Parisian counterpart, its facade has striking similarities: the grand rose, the two square towers at a similar height (66-69 m)... but while Paris stopped in 1345, Strasbourg kept going for almost a century, filling in the space between the towers, and adding a whopping octagonal spire on one side, reaching 142 m above ground.

Of course, there were plans to make the monumental facade symmetric, but the ground under the South tower wouldn't support the weight of 76 m of spire. In fact, huge structural repairs had to be made during the 19th century to avoid collapse.

The cathedral was the world's tallest building for a couple of centuries, from 1647 to 1874. Considering it was completed in 1439... Yeah, it didn't grow, it owed it title to the Pyramids of Giza shrinking from erosion and taller spires on other cathedrals burning down. Then it lost the title when churches in Hamburg, Rouen (another Notre-Dame Cathedral) and Köln were completed.

But talk of records is just talk, and 142 m is just a number, until you're faced with it. My favourite approach to the cathedral, to truly give it is awesome sense of scale, is the one I inadvertently took on my first proper visit to Strasbourg. From the North end of Place Gutenberg, walk along Rue des Hallebardes. The town's buildings will hide the cathedral from view for a moment, only for it to reappear suddenly at the turn of a corner, much closer, the spire truly towering over the surrounding buildings which also dwarf the viewer. I don't pass by there too often, to try to replicate the breathtaking reveal.

PS - We've already done a piece on the astronomical clock housed in the cathedral, an absolute treasure.

Sangaku Saturday #13

Last week, we uncovered this configuration which is also a solution to our "three circles in a triangle" problem, just not the one we were hoping for.

This is something that happens in all isosceles triangles. Draw the inscribed circle, with centre B, and the circle with centre C, tangent to the extended base (ON) and the side [SN] at the same point as the first circle is. Then it can be proved that the circle with centre A, whose diameter completes the height [SO] as our problem demands, is tangent to the circle with centre C.

But that's not what I'm going to concentrate on. Despite this plot twist, we are actually very close to getting what we want. What the above configuration means is that, returning to the initial scaled situation with SO = h = 1 and ON = b, we get

Knowing a solution to a degree 3 equation is extremely powerful, as we can factor the polynomial and leave a degree 2 equation, which has simple formulas for solutions. So, to finish off, can you:

1: prove that

2: solve the equation 2x²-(s-b)x-1 = 0, and deduce the general formulas for p, q and r that fit the configuration we are aiming for;

3: test the formulas for an equilateral triangle, in which s = 2b.

This last question is the one the sangaku tablet claims to solve.

Strasbourg's Christkindelsmärik

One of Europe's oldest Christmas market, and likely the most famous in France, is Strasbourg's. Its existence is attested as far back as 1570, appearing in the years following Protestant Reformation. Today it is a sprawling event, covering all the main squares of the central island of Strasbourg, and it's very busy, especially in the evenings and on weekends.

The traditional Alsatian name of the market is Christkindelsmärik, "the market of baby Jesus", while the city's more recent branding (since the 1990s) proclaims the town as "Capital of Christmas".

Pictures from 2018 - I haven't been to this year's market yet, but I plan to once my workload deflates - I get my annual stock of jams from the local producer's market!

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.

I'm a bit low on inspiration and time today (work starting to pile up), so here's a train in the snow from the recent trip to Mulhouse and Thann. The train itself is a bi-mode Regiolis B84500 set, waiting at Mulhouse as the Sun sets.

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.

First Christmas Market of 2024

Not as early as Ebisu, but still, just over a month before Christmas, I got to my first market on 23 November. On our way to Thann with two fellow hikers, our train was delayed and we had half an hour to kill in Mulhouse. I know the city centre is quite nice, so we went there, and found the Christmas market!

Place de la Réunion is gorgeous with its trompe-l'oeil facades (walls that have bricks, columns and other ornaments drawn on them) and church, which, unusually for a major French town, is a Protestant temple. It's extra-special with Christmas decorations, such as the town hall seen above. The water mill wheel is the emblem of Mulhouse, referencing the name's origin, Mülhausen, "mill house".

How early is too early?

Christmas markets have been a staple of the month of December in Alsace and Germany, and the concept of local specialities and gifts being sold in chalets has spread far and wide. Most are open for around a month, ending on Christmas Eve, maybe pushing a couple of days more.

Japan also has a few markets, and, considering how differently the date is celebrated (New Year is the family holiday), you'd think a Christmas market would be a little something to bring some cosy European atmos to wandering couples in the week or two running up to it.

Holy cow, the 5th of November! That is by a long way the earliest Christmas market I've ever seen! This was the one in Ebisu in 2016, just outside the Skywalk from the station (nowhere near as spectacular as the Mishima Skywalk), opposite a big mall. It was very calm, much less busy than the big shops nearby, which were also already decorated.

Sangaku Sunday #11

We're almost there! We have three relations between our unknowns, the radii p, q and r. Actually, let's write them in the general setting, with any height.

Set SO = h and ON = k (so the number b in the problem so far has been equal to k/h). Repeating what we've done in previous steps, and substituting q and r in the final equation so that we get an equation with just p (I've done it so you don't have to), this is what we're solving:

The plan is simple: get p with the last equation, deduce q then r with the first two. The execution of the plan... not so simple. That last equation is messy. Let's tidy it up a bit by noting that it is actually a polynomial equation of the variable x=squareroot(2p):

There are formulas for the solutions to an equation like this, but if we can avoid using them, we'll be happy.

Here's what I did - and you can do too: a numerical test. Let's take the simplest dimensions for a right triangle, h = 4 and k = 3. Replace in the last equation and notice an obvious solution. Deduce p, then q, then r. Jubilate - until you realise something is very, very wrong...

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.

Strasbourg's tramway's 30th anniversary!

On this day in 1994, Strasbourg inaugurated - or rather, resurrected - its tramway network. Like many cities in France, Strasbourg had a streetcar system until the late 1950s, when it was decided that cars would take over. 30 years of worsening congestion and pollution later, the town chose a tramway, which had made a successful return in the mid 1980s in Nantes and Grenoble, over an automatic metro to revitalise its transit service.

Unlike Nantes and Grenoble, Strasbourg looked to foreign streetcar manufacturers Socimi and ABB, who designed a fully low-floor tram with generous windows. The Eurotram was at first a 33-metre vehicle (original form seen above), which quickly proved insufficient. A lengthened version, with an extra motor module and carriage, appeared in the following years.

Personally, I quite like this tram for the massive windows, the very mechanical sounds as it runs, and the fact that the warning bell is a real bell (later models have an electronic bell which... just sounds worse). A downside I have noticed, though not for me specifically, is that it has a low ceiling.

After losing out in the 90s, national constructor Alstom won the next round of contracts for Strasbourg's trams in the 2000s. The Citadis model, fully low-floor and taller than the Eurotrams, entered service in 2005. More Citadis trams arrived in 2016, with a new design that I really like, and with special adaptations to allow it to run in Germany, as the network crossed the border to Kehl in 2017, a first for a French tram operator.

Today, the network consists of 6 lines, crisscrossing the city centre and heading out into the suburbs. A 7th line is in the planning stages, due to head North towards Bischheim and Schiltigheim. Despite refurbishment, the Eurotrams won't be around forever, and new trams are on order - more from Alstom.

Engelbourg: the castle with the hole!

A less glamorous variation of the 10 yen coin photo, but I did say back in March that this place made me crave Polo mints!

More seriously, I returned to Thann this weekend with two Japanese people working in France this year. The timing was impeccable, as there had been some snowfall in the previous days, giving us some spectacular views from the summit.

Also, just in time for the start of the Christmas market! Getting into the festive mood early this year... and no need for fake snow! At least, that was the case yesterday. Today, the weather got much warmer and it's probably all gone.

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.

The Mulhouse-Thann tram-train

Combining a suburban train service with the ability to navigate city streets sounds amazing. People can live nearer to the countryside, get frequent service into town, and, if everything lines up, commute straight into work without changes and avoiding the main station. The complementarity and opportunity to revitalise a branch line all sounds appealing... but a real challenge to implement. In France, only Mulhouse has truly achieved it.

Tram-trains aren't exactly rare in France: there are several lines around Paris, Nantes and Lyon have them (and many more had tram-train projects at some point). But, while the vehicles are capable of running in both modes, they are mostly used as a cheaper way to operate a line. The Nantes-Clisson and Nantes-Châteaubriant tram-trains, for example, which I have ridden, are just regional trains, running on heavy rail nearly all the way, and only stopping where the trains always used to.

Mulhouse is the only place in France to have true tram-train operations as described in the introduction: the tram-trains add traffic to line 3 between Mulhouse central station and Lutterbach, before switching to train mode and continuing on the branch line to Kruth as far as Thann.

The vehicles themselves are remarkable, as they need to be equipped for both streetcar and heavy rail operations, and each has its own requirements: lighting, horns, power supply, safety features... Mulhouse's vehicles are Siemens Avanto S70s, built in 2009-2010, and operated by SNCF as class U 25500. Similar units were introduced near Paris as early as 2005.

Fort National, Saint Malo

Work is starting to pile up on my end, so I have to make this a quick one.

This is Fort National, a building we saw in the post on MV Bretagne. It was built in the late 17th century by Vauban - one of many, many, many projects he designed for Louis XIV's grand plan to fortify the French border. It was called Fort Royal, a name which would stick for little more than a century, before the Revolution banished any mention of royalty. It became Fort Républicain, then Fort Impérial under Napoleon, and finally Fort National after Napoleon III's Empire was defeated by Prussia in 1870. This regular name changing was derided by a local nickname, "Fort Caméléon", but it also give a glimpse into France's political history.

The rock on which the fort sits, known as Îlette (the small island), has quite a sorry history. Before Vauban, it was apparently used as an execution ground by local lords, and during World War II, the occupying Nazis used it as a prison during Allied bombings - fully expecting the fort to be bombed.

Today, the fort is privately owned, but visits are allowed on occasions.

Saint Servan and the Solidor Tower

On the right-hand side of the Rance river, just before the fortified city of Saint Malo on the estuary, is the smaller town of Saint Servan. In fact, it technically isn't a town anymore, it was absorbed by Saint Malo in the 1960s. But for most of history, there was a stark contrast between the two, as Saint Malo fiercely proclaimed its autonomy several times. Hence the Solidor Tower.

Consisting of three tightly-bunched round towers and their connections, the Solidor was built in the 14th century by the Dukes of Brittany as a means to control the Rance estuary, against the rebellious Saint Malo if needed.

Like other fortresses, such as the Bastille in Paris or the towers at La Rochelle (another time maybe), its strategic value soon dwindled, and it seemed best-suited to serve as a prison or as storage during the late 18th-early 19th centuries. It has been an officially classified monument since 1886, and had housed a maritime-themed museum since 1970, though this appears to be in limbo and I can't find the tower's current function.

A walk along the coast on the West side of Saint Servan will reveal a bit more history: an old lifeboat station, a small tower in the sea that serves as a tide gauge... further up, a WWII memorial with the remains of concrete bunkers, and further along, a view of Saint Malo. It's a worthwhile detour for people visiting Saint Malo, especially if you're concerned that the city centre will be too crowded. But I think I remember parking here wasn't easy either; on a nice day, the locals who don't want the hassle of "intra-muros" would come here.

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

MV Bretagne's final crossing

I haven't got my eye on the Channel as much as I used to, so I only found out last weekend that this ship had its final run on the night of 3-4 November.

Bretagne was Brittany Ferries' first purpose-built cruise ferry, launched in Saint Nazaire in February 1989 and entering service in July of that year. At over 150 m in length, appointed with over 350 cabins and a higher level of comfort than other ferries in service at the time, she was designed to be the company's flagship, sailing the longest routes to Spain and Ireland.

As tourism between the UK and continent became more popular, Brittany Ferries' fleet of cruise ferries expanded further in the early 90s, to the point where Bretagne was no longer the company's largest ship by 1993. While Val de Loire took over the Portsmouth-Santander route, Bretagne became a regular on Portsmouth-St Malo, serving her namesake region. So, in the summer of 1994, it was she who carried my family over to new lives in France.

While not my favourite ferry, Bretagne is a particularly important one on a personal level. So it was nice to catch her by chance departing St Malo in July 2019, around her 30th anniversary. Five years later, and she would pass behind the islands off the Corsair City for the final time, bound for Le Havre to await her sale.

Kenavo, Bretagne!

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.

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 #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.

More faces on Dôtonbori

There are the classics on Dôtonbori: Kuidaore Tarô, there's a well-known animatronic crab, and of course the Glico man. I reckon this ramen-loving dragon is my personal favourite.

... though, giving it some thought, I quite like the hand holding a sushi and Spiderman reaching for a pearl too.

But oh boy, are you ready for some real randomness?

This is the front of the Dôtonbori Hotel. According to Atlas Obscura, the hotel had these pillars made to symbolise them welcoming people from all over the world - the faces represent East Asia, Africa, the Middle East and Europe -, at a time when domestic tourism was dwindling (early 1990s).

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!

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!

Sangaku Sunday #7

We're back with a new problem from Miminashi-yamaguchi-jinja! This is going to be more ambitious than the first one, though it won't be much harder from a geometry standpoint - the main tool will still be Pythagoras's theorem. But we really need to set the stage for this one.

Consider an isosceles triangle, with two circles whose diameters are on the height from the apex, tangent to each other, and so that the top circle passes through the apex and the bottom circle is tangent to the base. We seek to draw one more circle on either side, which is tangent to the first two, and tangent to two sides of the triangle.

Details and first questions below the cut.

The triangle is given: it is an isosceles triangle SNN'. For the sake of simplicity, let's shrink or blow up the figure so that the height SO is equal to 1 (for a configuration with height h, we will just need to multiply all the lengths by h). The length of the base NN' is therefore the fixed parameter of the problem, and, as the figure is symmetric with respect to SO, we only need to set the length ON as our parameter: set ON = b. Hence, we are working in the right triangle SON.

The problem involves finding the three circles that fit the configuration in SON. Let these circles have respective centres A, B and C, and respective radii p, q and r. The radii are the unknowns of our problem, and we need to find three independent relations between them to solve. From the sketch, it looks like there should be only one solution.

The first relation is obvious: 2*(p+q) = 1, as the diameters of the first two circles form the height SO. This is also very easy to solve: if we have p, then q = 1/2 - p.

A second relation must start to involve r. For this, project the centre of the third circle onto SO and ON, calling these projections P and Q respectively. Now we get to two questions for you to munch on.

1: Prove that

2: Get the lengths AC and PA. Deduce another expression for PC, and prove that

With that, we just need another equation to find p, and we'll be done.

You can do as you wish @todayintokyo, it depends on how much of a hint you want to give. ;)

Personally, I did tag the city for classification purposes, and I found that 3-4 lines of rambling tags can drown out the rest on the dashboard ("see more tags" appears). This doesn't work on the full-page blog site, though that might be customisable with some HTML knowledge.

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.

On the JR Nara line

Uji city and the the building on the 10-yen coin can be accessed by train from Kyôto by going roughly a third of the way to Nara. Other famous sites near the line are Fushimi Inari Taisha (Inari stop), and the studios of Kyoto Animation, famous for the music and sports anime K-On and Free! (Kohata stop).

The most recent type on the route is the 221 Series, and it's already getting on a bit, introduced in 1989. It won one of the Japan Railfan Club's two main new train design awards, the Laurel Prize, the following year. The 221 is used on the fastest Miyakokji Rapid services, which do the Kyôto to Nara run in under 45 minutes.

Green 103 Series sets can also be seen. This is the oldest type still in active JR service (if not, it's close), as it was introduced in 1963. In 2016, when I first visited Japan, I was living near Paris, and some Métro and suburban lines were running trains of a similar age, if not older, and these were atrocious in hot weather - no air conditioning, and ventilation only provided by opening windows! The RATP MP 59 used on Métro line 11 was stinky to boot; it was withdrawn just before the Games, no wonder! Point is, the 103 doesn't have air con either, but is at least trying...

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.