Another Short One Today, Just A Couple Of Christmas Decorations From Strasbourg. The "tree Of Cathedrals"

Running around Koblenz station the other evening, I couldn't resist the cute message on this local train. Instead of a deadpan "depot" or "not taking passengers", this train is "going to bed"!

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.

The Reality Checkpoint

In the middle of Parker's Piece (no relation to the Thunderbirds character) in Cambridge, stands a lamppost. The only lamppost on the common, a beacon and a reference for anyone walking there at night. A bit like the lamppost in Narnia. A lot like the lamppost in Narnia in fact, as it serves as a boundary marker between the university and the town, two worlds with distinct notions of reality. That's one theory behind the name, the Reality Checkpoint.

Another theory suggests one should use the checkpoint to assess their clarity when going back from the pub, and this view certainly calls reality into question... Not to worry, the Ferris Wheel is just being dismantled.

This ornate lamppost has been restored in recent years, and improved with a Dinky Door. Well, I say "improved", but the note on the door says "on holiday, please check reality yourself"!

Neuviller-la-Roche

Quick post today - just a few views of climbing out of the village of Neuviller-la-Roche in late March 2022.

Towers and trains at Oberwesel

Downstream from Bacharach and Kaub seen in the most recent posts, Oberwesel is a gorgeous town on the left-hand side of the Rhine, with many of its medieval walls and towers still standing. The railway was built alongside these walls near the river, and even goes between two towers, the Katzenturm (left) and Ochsenturm (right). Add the hills in the background, and it is certainly a spectacular train spot.

Here is another tower, the Haagsturm, in a view from the station platforms. (I just got off that train and failed to position myself in time to get the sign out of the way bottom left.) The two trains shown were the only types visible that day, as the intercity traffic was diverted to the other side of the river via Wiesbaden.

Further from the river, another section of town walls and towers runs through the hills. In the centre of the picture above, taken from the short but steep Elfenlay trail, is the Kuhhirtenturm (with raised drawbridge), with St Martin's Church rising behind it.

Oh alright, have a wider view from the Elfenlay.

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".

I had to do Eckwersheim village justice on the way to the canal and the memorial shown yesterday, and took the time to walk through it. As it is part of the Strasbourg Metropolitan Area, buses go there, and it's a rather nice ride through several villages, nearly all of which end in 'heim. And all these villages, within easy reach of the city, have at least a few streets with charming, timber-framed buildings, some richly decorated, with flowers and ornaments like the one above. An Alsatian village in rose season is quite the treat.

This building is evidently a restaurant, with the menus chalked up on the gate! I was there on a Monday morning but no menu on display - maybe it was too early, or maybe they weren't going to open because it was a national holiday - kind of. Pentecost in France got weird after 2003 and I don't fully understand it. Let's enjoy that door some more instead.

And it wouldn't be a lovely day in Alsace without storks!

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 #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.

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!