Remembering Eddie Jordan

A little local train in Hikone: Ômi Tetsudô is a private company that's been around in the area for over 125 years, hence the panel on this particular train, in retro colours.

Japan has many small lines run by small companies which were never nationalised. However, Ômi Tetsudô is owned by the larger private rail company Seibu, based in the North-Western sector of Tokyo, whose main route is Ikebukuro to Chichibu. Ômi mainly runs second-hand Seibu stock.

Summer highlight reel: Île de la Table Ronde

To the South of Lyon, the "Island of the Round Table" in the middle of the Rhône offers a fantastic escape from the city. While the East side is exposed to a lot of noise from motorway traffic, the inside and West shore are gorgeous, and the southernmost end is a nature preserve.

The river flows by at a steady pace, making it a good spot for a reaction ferry similar to the ones in Basel. Fair play to the locals, they thought that too!

Bridges now do the job - though the suspension bridge from Vernaison isn't doing too well. Built in 1959, it needs replacing and until then, traffic is limited on it so as not to overload it. This hasn't been helped by the North side of the island being an industrial estate.

In the centre of the island, one finds a ruined farm, the Ferme aux Loups. One thing @teamroquette likes to do is geocaching, and so we looked for some, but the most elusive geocache of all was the namesake of the island. There are pictures of a round table associated with the island on Google Maps, but we missed it. That said, one Google review also mentions that they couldn't find it, so who knows.

We did find these interesting and somewhat imposing water level meters though. Lay on them to measure yourself... and get the wrong answer!

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.)

Success for Aston Martin at the Spa 24 Hours courtesy of Comtoyou Racing! The British marque hadn't won the Belgian classic since 1948, and hadn't won a 24-hour race outright since 1959, which was the year the legendary Carroll Shelby won Le Mans with Roy Salvadori in a DBR1 (pictured above at Le Mans Classic in 2018).

Aston Martin have had success at Le Mans since, winning the GT class four times with the Prodrive-built DBR9 and Vantage GTE (pictured above at Le Mans in 2013, a tragic event for the team as Aston driver Allan Simonsen driver died in an accident early in the race). But in the races where GT3 cars are the headline, they have typically struggled to beat the powerhouse brands from Germany and Italy. Seen below is a predecessor of the new Spa winner: the 2013 V12 Vantage GT3, raced at the Nürburgring 24 Hours (pictured at Le Mans Classic in 2018).

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.

I can't remember riding a steam train before, though deep inside, I feel I probably had. Anyway, now I'm sure! This is the Chemin de Fer Touristique du Rhin, a short line near Colmar which runs steam engines and a set of old Austrian carriages, of which I'll say more in another post. Meanwhile, it's been a busy time for me recently, so this is just a few photos from the ride while I wind down.

Everything is ready for Tuesday! How this particular configuration works, as well as the one below, will be covered - we can talk about it on here too afterwards if anyone's interested.

C'est avec grand plaisir que je présenterai le mardi 16 avril à la Maison Universitaire France-Japon de Strasbourg une conférence sur la géométrie pendant la période d'Edo, avec en support le sangaku de Kashihara. Entre grande Histoire et petits calculs. Lien vers les détails 4月16日(火)、ストラスブール市の日仏大学会館に江戸時代の算額についてコンファレンスをします。楽しみにしています！ Looking forward to giving a conference on Edo-period geometry on 16 April at Strasbourg's French-Japanese Institute. Expect a few posts about Kashihara around then. Has it really been 6 years?...

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.

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.

Hikone Sawayama: base

Walking North along the railway from Hikone station, one reaches the base of the hiking trail up Sawayama. After passing Nagabayashi Inari-jinja, a typical shrine dedicated to the shintô deity of prosperity with its succession of red torii gates, several temples appear, featuring monuments to two historical figures of Hikone, Ishida Mitsunari and Ii Naomasa. More on them when we reach the top.

This is Ryôtan-ji Sanmon, the "gate to the mountain" which leads us to the grounds of Ryôtan temple and starting the short, sharp climb. As we begin, we are met with more popular Japanese deities: the Shichi-Fukujin, or Seven Lucky Gods.

Apparently Ryôtan-ji has a fantastic zen garden, but we missed it.