MiniWorld Lyon

Sangaku Saturday #13

Last week, we uncovered this configuration which is also a solution to our "three circles in a triangle" problem, just not the one we were hoping for.

This is something that happens in all isosceles triangles. Draw the inscribed circle, with centre B, and the circle with centre C, tangent to the extended base (ON) and the side [SN] at the same point as the first circle is. Then it can be proved that the circle with centre A, whose diameter completes the height [SO] as our problem demands, is tangent to the circle with centre C.

But that's not what I'm going to concentrate on. Despite this plot twist, we are actually very close to getting what we want. What the above configuration means is that, returning to the initial scaled situation with SO = h = 1 and ON = b, we get

Knowing a solution to a degree 3 equation is extremely powerful, as we can factor the polynomial and leave a degree 2 equation, which has simple formulas for solutions. So, to finish off, can you:

1: prove that

2: solve the equation 2x²-(s-b)x-1 = 0, and deduce the general formulas for p, q and r that fit the configuration we are aiming for;

3: test the formulas for an equilateral triangle, in which s = 2b.

This last question is the one the sangaku tablet claims to solve.

Feeling rather non-committal today, so here are some flowers from the Expo '70 Commemoration Park near Osaka. This is probably going to start a mini-series because the place is huge, so more information will come later.

Saint Servan and the Solidor Tower

On the right-hand side of the Rance river, just before the fortified city of Saint Malo on the estuary, is the smaller town of Saint Servan. In fact, it technically isn't a town anymore, it was absorbed by Saint Malo in the 1960s. But for most of history, there was a stark contrast between the two, as Saint Malo fiercely proclaimed its autonomy several times. Hence the Solidor Tower.

Consisting of three tightly-bunched round towers and their connections, the Solidor was built in the 14th century by the Dukes of Brittany as a means to control the Rance estuary, against the rebellious Saint Malo if needed.

Like other fortresses, such as the Bastille in Paris or the towers at La Rochelle (another time maybe), its strategic value soon dwindled, and it seemed best-suited to serve as a prison or as storage during the late 18th-early 19th centuries. It has been an officially classified monument since 1886, and had housed a maritime-themed museum since 1970, though this appears to be in limbo and I can't find the tower's current function.

A walk along the coast on the West side of Saint Servan will reveal a bit more history: an old lifeboat station, a small tower in the sea that serves as a tide gauge... further up, a WWII memorial with the remains of concrete bunkers, and further along, a view of Saint Malo. It's a worthwhile detour for people visiting Saint Malo, especially if you're concerned that the city centre will be too crowded. But I think I remember parking here wasn't easy either; on a nice day, the locals who don't want the hassle of "intra-muros" would come here.

"Those red cars were the bane of my existence" - 2000-2004 Ferraris

When Aidan Millward said something along those lines in a recent video, I felt it, because as a Häkkinen fan in the late 90s and early 2000s, Ferrari and Schumacher were the adversary. And like Aidan does in his videos, I have to admit that, looking back, they were only that insufferable because they were that good. The ultimate adversary of sorts, with generational talents in driving, managing and engineering under one roof: Michael Schumacher, Jean Todt, Ross Brawn, Rory Byrne, Nigel Stepney...

It took a few years and a few attempts, but in 1999, Ferrari won their first Constructors' championship since the early 80s, with Schumacher's first Drivers' title in red, and the first for Ferrari in 21 years, came a year later, with the F1-2000 pictured on the left. The momentum had built up, and for the next four years, the well-oiled machine that was the Scuderia with the aforementioned people in charge, were consistent contenders, always in the conversation to win races and championships... though more often that not, they were the only ones in the conversation. 2001, 2002 and 2004 were write-offs if you weren't Michael Schumacher. 2003 (F2003-GA on the right) was a closer affair, as a tyre war allowed McLaren and Williams to get close to breaking the streak.

But it didn't happen, and Ferrari won Drivers' and Constructors' doubles for five straight years, something no-one had done before, and only an equally dominant Mercedes has done since. All five cars were on display together at the Museo Ferrari in Maranello - the set-up may have changed, but I expect all five to still be shown in some capacity. After all, 2000 to 2004 were Ferrari's best years, even though, as someone who wasn't a Ferrari fan, they didn't feel like F1's best years.

Port du Crouesty

We start the new year where we left off in the last, on Presqu'île de Rhuys, but one year later. Or should that be two? The previous post was from the stormy 2013-2014 New Year celebrations, whereas this covers the calmer 2014-2015 change with most of the same friends, and our walk around the tip of the peninsula and Port du Crouesty.

These aren't the best photos given the relatively low light, but also this was before my current camera. Still, point it towards the sunset and the result isn't too bad.

Neuviller-la-Roche

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

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

Sangaku Saturday #4

In the previous info post, we went over the debate on the religious aspect of sangaku, and the fact that the absence of prayers on these tablets was more puzzling to some than the mathematics. As such, the tablets are not ema prayer tablets, but donations, which usually don't feature prayers on them. Case in point, some consecrated sake and French wine seen at Meiji-jingû in 2016.

Beyond wishing for good fortune and health, such donations serve two very worldly purposes: to contribute to the life and prestige of the shrine or temple (having a famous contributor makes the shrine famous by association), and to advertise the donor in return, as their name is on display. See this large torii at Fushimi Inari Taisha paid for by TV Asahi (テレビ朝日).

With that in mind, Meijizen's cynical comment from 1673 that sangaku aim "to celebrate the mathematical genius of their authors" may not far from the truth. The authors of sangaku are looking to gain notoriety through the publicity that the shrine or temple provides. But was the bemused Meijizen the target audience?

More on that in a couple of weeks. Below the cut is the solution to last week's problem.

The solution to the first problem (below the cut in this post) is the key. Name K, L and M the intersections of the three circles with the horizontal line. Then, by using that previous result,

Indeed, as in that problem, we can construct three right triangles, ABH, ACI and BCJ and apply Pythagoras's theorem in each.

Now, it suffices to note that KL = KM + LM, so

or, dividing by 2*squareroot(pqr), we get the desired result:

Inverting and squaring this yields the formula for r:

This gives us the means to construct this figure on paper using a compass and a marked ruler. Having chosen two radii p and q and constructed the two large circles (remember that AB=p+q) and a line tangent to both, placing M and C is done after calculating the lengths IK=CM=r and IC=KM=2*sqrt(pr).

Approaching 100 posts - yes I know, this one's adding to it, but with asks and reblogs possible on Tumblr, I consider the companion Instagram to be a more reliable publication counter, not to mention I have plans for content that will only be on Tumblr (sangaku solving). I also need to clean up some tags before the task gets too daunting, with the aim to get some major themes rolling and provide direct links to them in the menu bar.

Thanks for likes left so far, I hope you'll enjoy what's next!

Catène de Containers

To celebrate the 500th anniversary of the founding of the port on the right bank of the Seine estuary, Le Havre went big. They commissioned a sculpture from artist Vincent Ganivet... and he delivered a monument!

Standing at nearly 29 m tall, the arches are made with 36 shipping containers, representing Le Havre's half-millennium as an international trade hub. 21 in one and 15 in the other, they are arranged in a catenary shape which makes the structure self-supporting. There's stuff to satisfy a maths and physics buff in there somewhere... but I'll just concentrate on the fact that it looks cool, especially compared to its industrial and brutalist surroundings.

As a major port in Nazi-occupied France, Le Havre was bombed into oblivion by the Allies, hence most of the town centre's buildings were built at once in the late 1940s-early 1950s. The result is a very rigid, homogeneous, mineral urban environment, to which the Catène adds a welcome dash of colour.

But if nothing else (and we've established there is a lot else), it looks like it'd make a compelling Mario Kart track.