Another Quick One Today I'm Afraid, But Some Really Nice Memories From Just Over 4 Years Ago: A Proper

Sounds like a train, but isn't: the Saig to Titisee Rodelbahn

In fact, it's just a footpath. OpenStreetMap puts it at 1.2 km in length, and it's all downhill from Saig. A 165 m drop to be precise, which means an average gradient of nearly 14% - that is steeeep with four Es. In the winter, it should be covered in snow, and, with its four turns, you'd figure it would be a really cool route for tobogganing...

Well, that's what a Rodelbahn is, it's a sled/luge/toboggan track! If you look up the term, you'll come across summer Rodelbahns which are rides on rails (little roller coasters, I've seen one next to the Arzviller boat lift that I'll probably talk about one day, and the Bobbahn is a cracking bobsleigh-style ride at Europa-Park), but this is a natural Rodelbahn. Which runs on a hiking trail, so a few rules need to be laid out, such as pedestrians should hug the inside of the corners.

Most of the trail is in the forest, but once (if!) you reach the final stretch, the ride into Titisee with this view of the lake must feel incredible.

Notice on the left that the base of a pole has got some padding around it... That's not (just) for visibility in the snow! If you can zoom in that far, you might notice that the walls of the bridge at the bottom are padded too.

You'll probably be thinking "again!" once you've reached Titisee, but, as we joked with my sister, that "again!" won't come soon - you've got a long, steep climb back up to Saig first! Those numbers, 1.2 km and 14% gradient, aren't so amusing when starting from the bottom... And the train from Titisee to Schluchsee mentioned yesterday doesn't stop at Saig! XD

To keep or not to keep? The case of the Kyoto Imperial Palace

Completing the set of former Imperial palaces that I've visited, the question regarding Kyoto was a different one, when, after the Meiji Revolution, the throne of Japan was moved to Tokyo, ending Kyoto's 1000-year tenure as Imperial capital. The answer came swiftly: keep it. After all, it had only been rebuilt in 1855, fires having destroyed the palace multiple times in the Edo period alone.

The Shishinden throne hall, with its courtyard and traditional tachibana orange and sakura cherry trees either side of its grand staircase, was used for the coronations of Emperors Taishô and Shôwa (Hirohito) in 1915 and 1928, but since the accession of Emperor Akihito in 1989, coronations have been held at the active Imperial Palace in Tokyo.

Today, the palace is preserved and can be visited for free, albeit on a fixed tour route. Only the exterior of the buildings can be accessed, still allowing good views of the Shishinden and the gardens to the East of the palace.

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:

Ryôgoku Kokugikan

It's sumo season in Tokyo! Though, to be fair, it's sumo season three months per year, as this arena, the Kokugikan in Ryôgoku, on the East bank of the Sumida-gawa two stops from Akihabara on the Chûô-Sôbu line, hosts half of the year's six major tournaments.

It wasn't open when I visited Tokyo, so I didn't see inside, but even then, there are apparently, and unsurprisingly, a lot of sumo-related places to visit in Ryôgoku, like Ekô-in temple... and I missed them all! Well, apart from the odd statue.

I went to Ryôgoku area without a real plan, I wandered around and found some interesting places I'll get to another time, but one thing I think I planned was to try the "lunch of champions", the classic stew for sumo wrestlers: chankonabe, or chanko for short. It was rich and copious, as you'd expect, and, as I remember, I didn't quite finish it.

Here we are: Miminashi-yama

When I visited Kashihara, looking to explore some deep Japanese history in the former province of Yamato, I expected to move around a bit, but there was actually enough in Kashihara itself to make for a busy day.

First up was this curious green round space in the middle of a residential area on the town map I'd picked up. It just seemed conspicuous to me, I decided to check it out.

This is Miminashi-yama, one of the Yamato Sanzan, or Three Main Mountains of Yamato. Though it stood out on the map and it does stand out in the plain around it, it's not huge, and it's a short climb to the top where a shrine awaited.

In that shrine, a sangaku geometry tablet is displayed. By chance, based on a whim, I had found one! Nearly six years on, I've finally solved it - it's not very difficult mathematically, it's just taken me this long to get on with it, having said that, even today I'm still figuring out extra things on it! - and will be presenting it at a conference tomorrow. I wouldn't have thought it at the time... I guess curiosity didn't kill the cat that day!

Ruine Schauenburg

My hike in July took me to the South side of Oberkirch, while on the North side sit the ruins of Schauenburg castle. It's a short, but steep, climb from town centre (or you can drive up).

The castle was built at the end of the 11th century by the Duke of Swabia. It saw action mainly in disputes between local lords, most notably after much of the land around the castle was sold to the Margrave of Baden-Baden, around 35 km to the North. It became a ruin following a French invasion in 1689.

Today, a restaurant sits beside the ruin. When I was last there, there was a camera stand to take a clean selfie - you know, the one where you set a timer instead of holding at arm's length. That's where my photo on my professional website comes from.

As is the case from the South side, the castle has a good view of the Rhine plains, and Strasbourg cathedral sticks out. It would have stuck out even more back in the day, without the modern tower blocks. The lords of Schauenburg would have seen the massive gothic cathedral and its monumental spire being built... over the course of a few centuries.

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 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...

Schauinsland

Since it's been mentioned in the comments on the previous post, we might as well have a look at the Schauinsland, a 1284 m peak on the outskirts of Freiburg im Breisgau. Only 7 km from the Schwabentor, the base is accessible by bus, before boarding a 3.6 km gondola lift ot the summit.

Opened in 1930, the Schauinslandbahn was the first gondola lift built for continuous operation, with the cabins running through each end station at slow speed to turn around while letting passengers off and on. The cable car takes the riders up the hill, offering panoramic views of the Black Forest. And then there's the view from the tower at the summit... I visited in summer, but it must be fantastic in winter too.

The cable car is not the only transport infrastructure to be found on the mountain, as mines operated there until the mid-20th century, so some mining railways with preserved vehicles can be found. Some of the tunnels can be visited (though writing this post reminds me, I haven't been inside yet).

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.