After weeks of work, the S-Bahn is running properly once more! To celebrate, here’s one of my favourite little maths problems. It’s something I first noticed during the torturously long wait for a Merseyrail train before the timetable improvements. I haven’t a clue whether it has a real name, so for now, let’s call it the S-Bahn Paradox.
Suppose you want to go from Frankfurt West to Frankfurt Süd (this was the limit until the building work ended). Well, there’s a train precisely every five minutes during the day, and the journey takes almost exactly 15 minutes (more like 16, but let’s say 15). As you travel, you’ll pass trains travelling in the other direction. Supposing you leave just as the next train is coming in, how many will you meet along the way (including the ones at the start and end stations)?
15 minutes, a train every five minutes, that means that there are three five minute periods, and (not forgetting the train you meet at time zero), you meet four trains altogether, right?
Well, not quite.
Let’s do this the easy way, with pictures. Here’s our train line, straightened out and with the stations removed (as well as trains that don’t travel the full distance between West and Süd). Our train is on the left hand track, facing south. Each tick represents the distance that the train can travel in one minute. (It doesn’t matter that this distance may vary as the train speeds up and slows down – all that matters is average speed)
As you can see from the number on the right, we’ve just met one train, and there are three more waiting for us. The four theory’s looking pretty good right now. Let’s see what happens if we bump the clock along by one minute.
Hauptwache! It’s not just a smelly railway station!
Back in the days when Frankfurt was its own semi-independent Imperial city state, the Hauptwache and its partner the Konstablerwache (about five minutes walk down Zeil, Frankfurt’s main shopping street) were the centres of Frankfurt’s martial might. The army and police (not that there was any real difference in those days) worked from these buildings, and they also served as armouries. This meant that it was a target for revolutions, such as the uprising of 1833, and for enemy countries. When Prussia annexed Frankfurt, they shut down the Frankfurt army and converted Hauptwache into a jail.
These days, the Hauptwache (reconstructed after being bombed) is a pricey cafe. It’s normally surrounded by parasols, planters and benches, which make it rather hard to actually see the building. I’ve stripped these away, so you can hopefully see the lovely architecture better. And yes, there is really a suit of (carved) armour over the entrance, sitting on a pile of shields and spears – the one reminder of its time as an armoury.
Next on our whistlestop tour of Frankfurt in Lego, der Eiserner Steg!
The Eiserner Steg (Iron Bridge) is almost certainly Frankfurt’s most famous bridge – it even has a top 10 song named after it. Unfortunately, building it out of Lego makes it look rather chunky, compared to the real, rather elegant bridge. It probably needs to be constructed on a larger scale, with a more gentle slope on the arches.
Also, try as I might, there’s no way of crafting that Homeric Greek inscription in Lego…
Wow, long time no blog.
I recently discovered Lego Digital Designer and, immediately after, the Lego Digital Designer to POV-Ray Converter. With these two tools, you can create basically any Lego model you like, for free, without having to hunt for pieces, and then render a realistic image of it. So that’s neat.
So, here’s a model I created to try it out.
It’s a Frankfurt tram! To be specific, a Flexity Classic! These are a) mostly low-floor and b) bogied, which makes them a bit of a pain to model within the limits of Lego Train sets, since these are all designed around high-floor mainline stock. I’ve cheated, and attached the wheels straight to the body. Hopefully this way of doing it looks OK. Other caveats: the doors are the wrong colour (they should be azure, which is the closest I could find to the Frankfurt Straßenbahn green in Lego, but there are no azure window pieces. Luckily, the window frames are meant to be black) and the articulation is rigid. Although you can’t see it here, I actually went to the effort of making sure all the transformers and air conditioners on the roof were accurate. Front probably needs some work too.
Other than that, there’s nothing too complicated about it. I thought the wing mirrors would be tricky, but modified 1 x 4 offset plates (4590) work pretty nicely!
Stay tuned for more Frankfurt-y Lego things…
I wrote this poem a while ago, and am moving it from my DeviantArt account to here for easy finding on request (!) This is a paradelle. Read the link! Paradelles are great and everyone should try them.
(A request! A real unfished-for request! Best Christmas present ever!)
The 90s was the decade when everything sped up! Sure, in the previous decades, technology and entertainment and politics changed, but that was so slow. The explosion of culture in the last decade of the century was completely unheard of before, and completely reshaped the world. The 90s produced so much new stuff, and got rid of so much old stuff, that only those who spent their childhoods in the 90s will ever truly understand that revolution! Already, it’s an era we look back on fondly, and with nostalgia. That’s why we at Sparrow & Dove have put together a list of 7 things, from every part of life, that only 90s kids will ever truly get!