Success For Aston Martin At The Spa 24 Hours Courtesy Of Comtoyou Racing! The British Marque Hadn't Won

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.

Shionomisaki: Honshu's southernmost point

Southern Kansai is probably best known for the Kumano Kodô, the pilgrimage routes related to the Nachi shrine and temple complex with one of Japan's most famous waterfalls. But Southern Kansai is also Southernmost Kansai, and the Shionomisaki peninsula, part of the town of Kushimoto is as South as it gets for the main island of Japan.

Beyond this lawn and those rocks is the Pacific Ocean: just water for thousands of kilometres in this direction! The nearest substantial landmass due South from here is Papua New Guinea, 35° latitude lower, or 10% of the Earth's circumference!

That's a nice lawn by the way, it would be a shame if someone...

Shionomisaki is basically Honshu's version of "Land's End" or "Finistere", and there is a little leisure complex to mark it. The recent Geopark Centre is good, but the blue buildings in the previous shot, the shop and restaurant and the observation tower, are from a different time and showing their age - case in point, the children's cups in the restaurant with the mascot of the 1998 World Cup in France!

Sangaku Saturday #1

In this series, I'll try to put something up every Saturday or Sunday (depending on my plans and/or your time zone), either with a puzzle, a solution or some historical info on sangaku. If people find this interesting (and editing maths on this probably will be), we can work step by step to solving full sangaku problems.

Let's start with a puzzle, one of the most basic tools of the genre. Two circles, with radii r1 and r2, are tangent to each other and tangent to a same line.

This means that the distance between C1 and C2 is equal to r1+r2, the distance between C1 and A is r1, and the distance between C2 and B is r2. Moreover, the lines (AC1) and (AB) are perpendicular, and the lines (BC2) and (AB) are perpendicular.

Can you prove that the distance between A and B is equal to

???

Hints below the cut, questions possible in the reply section.

Take your time, I intend to show the solution next weekend.

Hints. Place the point H on the line between A and C1 so that the distance between A and H is equal to r2. Then the triangle HC1C2 is a right triangle with two sides of known length. A famous theorem gets the third length, which is equal to AB.

It's time to go back to Kashihara, and let's start by meeting the local animals!

I'm getting real "fancy pants" vibes from the cat! But to be fair, it is a darn good looking cat.

That is all until I think of something more intricate to talk about.

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.

The CFTR's steam train

In a serious rain shower, the Chemin de Fer Touristique du Rhin's train stops at Volgelsheim station, where the association that maintains the line has its museum. The train itself is made up of former Austrian carriages built in the 1920s with what I suspect were 2nd and 3rd class seating.

The locomotive is a T3 tender built around 1900 at Graffenstaden, just South of Strasbourg, for the Alsace-Lorraine Railways. At the time, the region was under Imperial German control, hence the Eagle logo and German inscription "Elsaß-Lothringen" above the number. The association has two of these, nicknamed Berthold and Theodor. These are supported by small Diesel engines; on our trip, one of these hauled the train to the depot, where the extent of the association's work is on display. The active engines are maintained here, while others are being restored.

Peut-être un jour? - To run again one day?

The town of Breisach, on the other side of the Rhine and therefore in Germany, is visible, and a boat carries passengers across the river from near the depot.

Schwarzwaldpokal!

Finally, as someone who used to watch a lot of it, I've been to a ski jumping event in person! And it wasn't just jumping, it was nordic combined, which sees a jump followed by cross-country skiing; the same people, doing both. It happened today at Schonach, in the Black Forest, next to Triberg and all those cuckoo clocks.

The hill was built in October 1924, and is therefore celebrating its 100th anniversary, hence the big "100" on the knoll. Much younger is the concept of the Womens' Nordic Combined World Cup - it only exists since 2020! Ski jumping has been the final frontier for women in winter sport, so it was great to see men and women compete.

The longest jump of the day was 102 m, twice, for Norway's Jarl Magnus Riiber and Austria's Thomas Rettenegger. In the womens' contest, Germany's Maria Gerboth took the lead with 95 m.

The competitions follow the Gundersen format, in which the jump comes first, and the gaps in points after that equate to time handicaps at the start of the cross-country race. Above left are the time penalties for some in the womens' race, and right, the start, with the top three starting almost together, while the others must wait. The race is effectively a pursuit, with the athlete crossing the line first being the winner.

In the end, Norway were unbeatable today. Despite starting with a 25-second handicap, Ida Marie Hagen took the lead of the womens' 4 km race in the middle of the first lap and didn't look back, celebrating her upcoming win in the final corner (left picture)! Nathalie Armbruster got the host nation's only podium of the day, ahead of the Kasai twins from Japan.

The mens' race was longer, at 10 km, and required more effort management. On the final lap, Jarl Magnus Riiber lost his long-held lead to Jens Luras Oftebro (right picture), who would defeat Austria's Johannes Lamparter in the final sprint to the finish.

All in all, a great day. Fresh in the morning, especially with the hill in shade, which also made photos more difficult, nice in the sunshine in the afternoon, the races were easy to follow (it was surprising how much of the cross-country course we could see), the atmosphere was really pleasant, and I wasn't going it alone for a change!

I sighted this Vectron at Karlsruhe: a Dual Mode, which can move either with its Diesel engine or getting its energy from the overhead wires (German 15 kV AC only for now). However, it only delivers a third of the power of a standard all-electric Vectron, and is therefore not designed for main line hauling, and is expected to be more at home near sorting yards. Deutsche Bahn have also bought some of these as ICE rescue locomotives, serving when a train breaks down.

For Vectron!

Produced since 2010 by Siemens, the Vectron is a modular locomotive platform with various engine options - AC electric, quad-voltage for use across Europe, "last-mile Diesel" option for parking, Diesel motors, dual mode/hybrid... It hauls both freight and passenger trains. But the main reason I've wanted to mention the Vectron is...

this Mitchell and Webb sketch!

This is from series 3 of That Mitchell and Webb Look, which was aired in 2009. The Siemens Vectron was officially launched in 2010, so it's fair to say that the name appearing in both is a coincidence. However, when I see a Vectron, it reminds me of this sketch, so it's harder for me to take this train seriously!

But it is serious business, as it is one of the most common locos in continental Europe. Only Iberia (due to using a different gauge) and France (because if it ain't Alstom, they'll oust 'em) don't see much of them. The examples shown here are from Germany, Switzerland and Slovakia, and were all pictured in the same area of Germany. The quad-voltage version in particular allows companies to carry freight all over Europe, they're virtually borderless.

Yet here I am, still snickering at the name, by Vectron's beard!

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.

The train to Kushimoto: a JR West 283-series Kuroshio express. The sets are getting on a bit, they were introduced in 1996 and the livery could do with a refresh, but the "dolphin nose" is distinctive, and I, for one, really like it. They're also quite rare, only 4 sets exist.

The Kisei Main Line is essentially the coastal route from Nagoya to Wakayama. The full trip around the Kii peninsula takes 8 hours by express train, with a change required at Shingû or Katsuura, but it's definitely scenic as it gets very close to the sea (photo between Kii-Tahara and Koza).