Grand Geroldseck Castle - Only A Cat Of Different Coat

Kanazawa-Tsuruga Shinkansen extension open!

Since yesterday, these beauties (seen at Takasaki on the same trip I had that "race" into Omiya) have another 120 km of track to play on, as the Hokuriku Shinkansen extends further West along the coast of the Sea of Japan into Fukui Prefecture.

Of course, the best news here is that travel times between Kanazawa and Tsuruga are slashed - let me rephrase: halved - compared to the previous fastest express services. The dream of completing the route to Kyoto and Osaka is in reach, and if you add the Maglev line, there could, in the long-term future, be three full high-speed Tokyo-Osaka lines: the historic South coast route, the scenic North coast route and the ultra-fast route straight through the middle.

But there are other consequences. As has become the standard along the Hokuriku route, the old line has immediately been sold off to a "third sector" company - largely run and subsidised by local authorities for as long as they're happy to keep the line open. Only all-stop trains are operated by these third sector companies, so there are only two options: very slow local trains, or very fast, but all the more expensive, high-speed trains. No rapids, no expresses.

The express trains which used to go to Kanazawa now all terminate early at Tsuruga, including the Thunderbirds - of course, technologically advanced Japan has more than the five Thunderbirds Gerry Anderson could muster! This display board seen in 2016 is not likely to be seen again. And if the route to Kyoto is completed, will the name disappear altogether, or continue as an omnibus Shinkansen service to Toyama? Maybe resurrect the original name Raichô (yes, similar to the Pokémon)?

Train geek notes aside, the future's hopefully bright for the region this new stretch of line serves, which was hit hard by the New Year Earthquake.

北陸新幹線おめでとうございます!

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 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.

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.

Remembering Eddie Jordan

I started watching Formula 1 properly around 1998, which meant that my introduction to the Jordan Grand Prix team, which made its top-class debut in 1991, was with the awesome yellow and black "hornet" liveries. The squad mounted an unlikely title challenge with Heinz-Harald Frentzen in 1999, and, while I was more of a Häkkinen fan at the time, if Frentzen had done it, I'd've been chuffed too.

Unfortunately, I cannot remember seeing any of these "hornet" cars in person. But I have seen a couple of Jordan's cars up close: a Honda-powered EJ12 at the Honda Collection Hall at Motegi (1st picture), and the second, at the private collection of the Manoir de l'Automobile in Brittany. It is painted in the team's 1997 "snake" livery, but it's not a 1997 car (the airbox out of shot is wrong). It has a high nose, which Jordan started using in 1996, so it could be a repainted demo car recycling the 1996 type, but then the sidepods are wrong! I think it's a 1995 car, with a 1996 nose, in the 1997 colours. What a mess!

Ultimately, Jordan was a midfield team that allowed good drivers to beat the front runners from time to time, and that, for one year, looked like it could morph into a top squad itself. Alas, that didn't transpire, but I will remember the yellow cars very fondly.

Cheers, Eddie!

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.

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.

We've seen these trains before, quite recently in fact, but they're back, on a much brighter day to really make their colours pop, and in a different border station between France and Germany.

This is Lauterbourg, the easternmost town in France, in the North-East corner of Alsace, and it shows the contrast between the line on the left-hand side of the Rhine, and the one on the right-hand side. The line from Strasbourg to Wörth am Rhein is not electrified to this day, and only sees local regional traffic. Nonetheless, Lauterbourg appeared to have a massive yard back in the day, now just a flat expanse of disused rails.

At the North end of Lauterbourg station, we find some old German mechanical signals, still in use!

Finally, like in the previous post on these trains, I have an amusing place name to share. It's more funny to pronounce with an English accent than anything else, but it also looks like a game of Countdown gone horribly wrong!

The best timepiece in the world (IMO): Strasbourg Cathedral's Astronomical Clock

I could go on about this thing for ages. There's so much history, so many symbols to spot, and so much information on display... This is going to be a long one.

I guess I'll start with the artistic aspect on which I have the least to say because it's the least up my alley. There's loads of mythology and Christian symbolism going on on this 18-metre tall monument, and these are the main draw for the general public, because they move around.

Like cuckoo clocks in neighbouring Schwarzwald, this astronomical clock has automatons. Every quarter hour, the lower level of the photo above sees a change of "age": a child, a young man, an adult and an old man take turns to be in the presence of Death, whose bells toll on the hour. At high noon, the upper level also moves, with the 12 disciples passing before Christ, and the rooster at the very top crows.

Moving on to what really makes me tick: the amount of information on this clock is incredible. The time, obviously, but actually two times are on display on the clock at the bottom of the picture above: solar time and official time. Given Strasbourg's position in the time zone, there is a 30-minute discrepancy between the two. Then there's all the astronomical stuff, like the phase of the Moon (just visible at the top), the position of the planets relative to the Sun (middle of the picture), a celestial globe at the base (pictures below, on the right)...

The main feature behind the celestial globe is another clock displaying solar time, with the position of the Sun and Moon (with phases) relative to the Earth, sunrise and sunset times, surrounded by a yearly calendar dial. These have remarkable features, such as the Moon hand that extends and retracts, making eclipses noticeable, and the calendar has a small dial that automatically turns to place the date of Easter at the start of each year. This sounds easy, but look up the definition of Easter and note that this clock is mechanical, no electronic calculating power involved! Either side of the base, the "Ecclesiastic Computer" and the "Solar and Lunar Equations" modules work the gears behind these features.

The accuracy of this clock and its ambition for durability are truly remarkable. Relative to modern atomic time, it would only need adjusting by 1 second every 160 years, and it correctly manages leap years (which is not as simple as "every 4 years"). It just needs winding up once a week.

Finally, the history. The monumental clock was built in the 16th century, and used the calendar dial above, now an exhibit in Strasbourg's city history museum. It slowly degraded until the mid-19th century, when Jean-Baptiste Schwilgué restored the base and upgraded the mechanisms. The "dartboard" on the old dial contained information like the date of Easter, whether it is a leap year, which day of the week the 1st January is... - all of which had to be calculated by hand before the dial was installed! - and was replaced by the Ecclesiastic Computer, which freed up the centre space for the big 24-hour clock, complete with Solar and Lunar Equations.

As you may have gathered, I am a massive fan of this clock. Of course, nowadays, all the imagery and information would easily fit into a smart watch, but a smart watch isn't 18 metres tall and powered by gravity and gears!

Izumo Taisha's rabbits

@shoku-and-awe made a great post on the rabbit statues at Izumo Taisha and why they're there, so I'll only add that they are all over the shrine's grounds, and as far as East as the Ancient Izumo History Museum.

In the gardens, the rabbits are depicted doing all kinds of things: reading a book, taking pictures, birdwatching... Yes, all that!

And of course, there are a lot of rabbits facing the shrine buildings and praying.

The plaque behind these two recognises Senge Takamasa and Kunimaro, father and son, current and presumed future chief priest of Izumo Taisha. Tracing their origins back to the rulers of the Izumo province way back in the Nara period (Takamasa is the 84th head of the clan), the aristocratic-priestly Senge family has very much stayed in high society to this day, from being involved in politics and governor of Tokyo around 1900 (the shrine had been taken out of their control following the Meiji revolution and the abolition of the nobility), to Kunimaro marrying an Imperial princess (who no longer holds the title as per the rules) in 2014.