Sangaku Saturday/Sunday Is Taking A Week Off.

Can't read my, can't read my, No you can't read my 岡 face!

A ceux qui pensent "on n'a jamais essayé": si si, on a essayé. Et ceux qui en sortent nous laissent assez invariablement les mêmes messages.

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.

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.

A few bugs seen this summer!

The East side of Île de la Table Ronde is the noisy side, with the motorway running on the opposite side of the river. It's also the side with the widest walking track, and it was teeming with grasshoppers! I noticed a couple had landed on my shirt and were just chilling (maybe interested in eating the fabric? I'm thinking of locusts, aren't I). @teamroquette took the photos as I was unsure how much I could move without them jumping off. Turns out they were really chill.

Other critters seen this summer include some seen at Oberkirch: bees working on flowers, that's a classic, and... some less fortunate insects.

Bonus - posted here because my sister will kill me if I post this on Instagram where she can see it: a spider hanging out by Nonnenmattweiher lake.

A little local train in Hikone: Ômi Tetsudô is a private company that's been around in the area for over 125 years, hence the panel on this particular train, in retro colours.

Japan has many small lines run by small companies which were never nationalised. However, Ômi Tetsudô is owned by the larger private rail company Seibu, based in the North-Western sector of Tokyo, whose main route is Ikebukuro to Chichibu. Ômi mainly runs second-hand Seibu stock.

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!

Up the Monumental Stairs at Auch

Some 75 km West of Toulouse, the city of Auch is far away enough for red brick to be far less prevalent in buildings. It developed along the Gers river, with the higher-ups living... well, higher up.

The Monumental Stairs were built in the 1850s when, following a rebellion against Napoleon III's coup installing the Second French Empire, the prefect decided to give the townsfolk something to do (per the city council, "créer des chantiers afin de donner de l'ouvrage à ceux qui en manquent"), rather than just repress. The results were a water and gas distribution network, and the Monumental Stairs, creating a comfortable link between the upper town and the riverside 35 metres below. Later, a statue of d'Artagnan, a musketeer made famous by Alexandre Dumas novels, was added.

Behind d'Artagnan here, rises the Tour d'Armagnac, a 14th-century prison. Unfortunately, it is privately owned and cannot be visited, unlike the neighbouring cathedral, built between 1489 and 1680.

While the back of the cathedral, visible in the top photo, is clearly gothic, which fits the start of construction, the front facade is in a later, classical style. This would fit the timeline, as cathedral building usually started with the crypt and the altar, working outwards, and finishing with the massive entrance and towers. Walking away, further West, we encounter one more figure of the town: Intendant Mégret d'Étigny, who administrated the Auch-Pau area under King Louis XV, and is credited with infrastructure improvements in the region at the time.

Suzuka Circuit

April became Formula 1 month for Suzuka last year, but back in 2018, it was the venue for a Super GT test weekend. I took the opportunity to enjoy the circuit in a much calmer atmosphere, though the weather was very changeable on the day (I got drenched walking down to Inô station afterwards!).

Opened in 1962, the track is very much enjoyed by drivers for its large variety of corners (the Esses, hairpin, chicane, the double-apex sweeper at Spoon, the dauntingly fast 130R...), and has an unusual figure-of-8 shape, crossing over itself - the bridge is before this hairpin, just out of shot on the left. On a test day like this, most grandstands are open to all, offering the viewer a similar variety of angles and vantage points.

The paddock was also open for the relatively few fans to walk around and have a chance to meet the drivers (James Rossiter pictured above). But even then, 2018 was the year Jenson Button was racing, and his garage got a lot of attention!

Despite the weather which turned miserable in the afternoon, the teams didn't shy away from running. There were even a couple of Safety Car training moments: rolling restart, and full SC procedure with class separation.

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...