How to make railway timetables (with graphs) - Numberphile
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- čas přidán 20. 11. 2021
- Featuring Hannah Fry discussing Train Graphs.
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Hannah is the best!
Do we know if the same type of graph is used in any instance other than transport time tabling?
Thank you! I've always been fascinated by the graph of Napoleon's march into Russia, and the devastating return, for the elegance with which so many factors are displayed in a single coherent form.
Here, you have shown me a wonderful tool for detecting and preventing collisions on any system where a single line must service multiple "users" without any coordination but reliable clocks and speeds.
@@EdwinWiles does falling rain interest you?
Because any idea can be interesting if enough information is considered!
Hello World is awesome. I'll sure check out this one too! =D
I am a train planner in the UK and we still use train graphs even in modern day software. They are incredibly useful and I don't know what I'd do with out them.
Same in Sweden. The graph is digital, but it's the same graph.
@Autumn Smartphone What happens on the day is none of my business 😂
@Jan Silas I believe ours have the time on the X axis and locations on the Y Axis. They are very similar to the one in the video at 7:15 in the bottom left.
And standardized time... at known markers you can speed up/slow down a bit to keep pace with the schedule.
@@cameronl59201 Kinda literally is your business, being that you're a professional train planner. :P
I saw something on a TV programme about these graphs before a few years ago. These graphs result in timetables that are completely functional and safe and trains arrive at the right place and the right time to pass each other safely without having to wait around too long... but apparently also the timetables that result are very conservative with unnecessarily big gaps between trains which reduces the efficiency of the network. A more intelligent system which is dynamically adaptable could make better use of the limited resources of tracks and rolling stock. It's something AI could probably do well. I'm sure that segment of that programme was a massive oversimplification though.
But then again they tried to overhaul train timetables to improve things a few years ago and there was a massive public backlash seemingly just because people had to change their routines slightly. I never had any problem with the actual new timetables.... well no more problems than the previous incredibly bad ones.
Hannah recently recovered from cancer treatment. Glad to see she is doing well.
As a french man, I'm personnally offended that you didn't include Croissant on these train tracks.
@@robbiepmusic Ah sí, pero yo prefiero pan de chocolate.
How long again does it take to get from Croissant to Alain Prost?
The french debate is not really between "croissant" or "pain au chocolat", it's rather between "pain au chocolat" or "chocolatine" (yet these designate the exact same 'viennoiserie')...
Croissant is actually on the other line, the one from Paris to Orleans. The station is right between "Hon Hon Hon" and Soup du Jour.
Sounds kinda flakey...
Fun fact, this is also one of the first practical uses of a "Space-time" diagram, far before relativistic applications. Simply swap the axes, make it photons not trains, and you get a Minowski diagram.
I'm wondering about what type of application 🤔 can use the negative values? Instead of time.
Yep, those are worldlines!
Glad I wasn’t the only one thinking of a Feynman diagram
Wasn't one of the things that inspired Einstein train stations, railways, and their clocks?
@@nmpspiky that depends greatly on what you mean by negative values. Sometimes, things which seem mysterious and confusing actually turn out to be obvious once you take the time to clarify your thoughts.
In Japanese, the timetable is called ダイヤ (daiya) which is short for diamond because of the shapes of these plots. This all came together for me when I asked station staff in Nara for the train to Kyoto and they pulled out a book filled with these graphs for all lines in the network. What a beautiful way to display the network, I thought.
I love the way Prof. Fry lights up enthusiastically about the mention of gruesome train accidents.
A tad concerning in the best way.
Perhaps she's a fan of the WTYP podcast :)
Train Crash Corner
@@pvanb on what podcast I dare ask you?!
*Professor Fry
With cutting costs, the SNCF shut down so many station. I'm so sad I'll never visit Alain Prost, Zinédine Zidane and Ohlala on my way to Normandie !
"Let's meet at Ohlala at 5pm" 🇨🇵🥖
At least we'll always have Paris.
I was looking forward to visiting Croissant and Déjà Vu as well. Sadly, this will never happen.
At least you can still see the Eiffel tower, right before you arrive in Rouen
Baguette is still up and running, then? That's a relief.
Hannah is adorable, it’s always fun seeing someone getting this excited about something they’re passionate about
My dad taught me about loop lines when I was small. On single track railways there was a "loop" on a hook. If the loop was there then you took it and you had control of the the single track. If the loop was not there you had to wait in a siding for the train coming the other way to give you the loop. At the other end you hung up the loop so that a train going in the opposite direction could then take possession of the loop and any train having possession of the loop would know that they had control of that section of line. Now 75 (retired from programming at 72) I recall with gratitude to my dad the number of times I thought about it as I set a flag for a bit of code that didn't want to meet another bit of code coming in the other direction. 😀
Thanks for the explanation regarding using a “loop” to decide whether a train can continue or wait for the opposing traffic.
If you're interested in train time tables, there is way more to discover, like the Swiss concept of "Integrierter Taktfahrplan" or "integrated regular schedule". It's more or less how their complete train system works nowadays. And it is so successful, that Austria has adopted it as well and Germany is in the process of adopting it.
This concept doesn't apply to a single track, but to the whole train network. And it has also some pretty nice mathematics behind it as well.
That will never work in the UK, because you can't work it out on brown paper with a sharpie. 😉
"... and Germany is in the process of adopting it."
And this process will only require another 50 years. Hooray Deutsche Bahn.
@@lonestarr1490 And it will, like everything Deutsche Bahn, be late 🤣
@@blindleader42 trust me, that would be possible. You just need a big sheet of brown paper. ^^
@@jullit31 Ahh, but will it ever after tick along smoothly with detailed instructions to handle any situation, mandatory training, yearly refreshers, and an attention to order second to none?
2:41 I am French and I am rofling at the names X'D
Especially the accurate observation that "these are getting increasingly problematic as we go along" XD
As a math-teacher I love this episode, because of it’s simplicity to express and visualize a real and complex problem, that gives a lot of meaning to graphs.
My math teacher was called Søren Bruun! But not Jensen, so it can't be you. Must be your namesake :)
I used to work at the Children's railways in Budapest and we used sort of the same thing to model our timetables. Whenever I pulled out something similar in front of the passengers they were sooo confused, little did they know it was actually pretty simple!
hűvösvölgy?
@@amatsird-nais7604 igen / yes
That's a fantastic little railway.
Similar graphs are used by traffic engineers to help "platoons" (groups of cars) going in both directions get a green light at the same time, just like trains get passing sidings at the same time.
I wish they did a better job. On Sunnyside Rd in Clackamas, OR every "platoon" is forced to stop at every light at every signal for the 10 miles of that Blvd. Same thing for all intersecting streets. HUGE waste of fuel.
But then again, Noble St in Fairbanks AK... hit a green light, you will hit all greens.
Fifth and Sixth Avenue in Anchorage also have very long green waves, it’s magical going through town on them.
@@petergreenson I know, it feels like you're in a parade! In Fbks it was set up like that to combat the winter smog issue from idling engines at stop lights and the cold smoggy air layer on the ground. Boy I miss winter in AK's interior!
We (uncreatively) call these diagrams time-space diagrams
Came here to say the same!
What a surprise ! I worked for a couple of years on the french software for this.
We restricted the scale to big stations (instead of whole lines) and they still had some of these few meter long paper versions of it.
Over the years, functionnalities and constraint added up. Like the train needs to stop at least x minutes at a platform for the passengers to have to time to get in or out. Pretty obvious stuff you'd think. But then there are cases where a train has to leave by the same side it came in. Extra time needed for the train driver (pilot?) to go from one end to the other. Cases where 2 trains make a bunch of the road attached together, and need to separate. So on and so on.
I remember a day when I spotted a bug : when trains where crossing at a specific type of intersection, from the exact same beginning half minute to the exact same end half minute, the software wouldn't compute it as a problem.
I remember the feeling. How long did the soft work with this bug ? No idea. Did it cause any trouble ? No idea. WHAT HAVE WE DONE !?*
* That was for dramatic tension. Obviously it wouldn't result in a collision, train drivers are better that than, but it could be the pebble in the planning like that rogue train spoken of in the video.
Whatever the timetable said, the signalling system *should* prevent a collision, and that has been the case since the 1840's approximately. Of course, ideally the timetable shouldn't present the signalling system with such a case. But trains running a little off the timetable, such that the signalling does hold a later train from making a conflicting movement - it happens all the time.
I can only imagine what that must've felt like! Was it a simple case of >= instead of >, or something more complex?
@@ximono It was obscured by addition of variables on both sides of the condition, but it boils down to it yes.
@@cr10001 there have been accidents though where the timetable had the trains cross at a point where they couldn’t, so one of the trains always had to be a little late or a little early. And one day they weren’t and at the same time there was a fault at the signalling system. So yeah, it can be one factor contributing to an accident.
@@XramDivad I would say it was almost never the case, that both trains would arrive at the crossing point at the same time. Generally, the first train to arrive would wait for the other. The two most prominent examples of single-line British collisions that come to mind (can't speak for US), Norwich Thorpe and Abermule, were both where one train was (or was presumed to be) running late so the other train was 'sent on'.
Dr. Fry's voice & accent is done of the best ASMR I've ever heard! 😍 I'm melting!
Great to see Dr. Fry again here on Numberphile--her enthusiasm was one of the things that originally attracted me to this channel.
Me too...
Prof Fry..
"attracted" - of course! ;)
enthusiasm, right :-)
"Enthusiasm" come on :D
I vote that we replace the word "rogue" with "naughty" in all well known expressions. For example, astronomers believe the galaxy could contain billions of naughty planets
More Hannah Fry please!
Tufte would love this. Amazing how some seemingly impossible problems become solvable with the right visualization.
He actually used this type of diagram on the cover of "The Visual Display of Quantitative Information".
@@GeorgeKlucsarits Unfortunately, I lent out my copies and never saw them again 🤦
In dunno why but dr. Fry makes Math so interesting and fun to listen too!
2 mins in, and I'm loving the gung ho attitude of these ancient people
Ah yes, the people behind the Roman steam trains. I've heard Julius Caesar was a big fan. ;)
The 19th century really was a different country
I'm French and I just wanted to say that I love those train station names! Great video!
I love how this shows how important having dedicated tracks for fast trains (high-speed expresses or non-stop freights) and slow trains (regional trains, slow freights) is. If you have only one set of tracks (or even just one for both directions), you can run fewer trains in total, as slow trains will always be in the way of faster ones. If there's dedicated tracks, you can just have a lot of slow *and* fast ones behind each other. In my region they ripped up a lot of tracks, so now the remaining ones are really full with a mix of trains. That leads to a worse timetable for everyone :(
And let's not even mention what the UK Government did on Thursday!
Sidings, sidings everywhere! ... to the point where it can be an extra track anyway.
@@mastertrams What DID the UK government do on Thursday?
@@DrewLevitt cancel the Eastern leg of HS2, thus getting rid of the dedicated high speed line that would have freed up the capacity on two major mainline railways (the Midland Mainline and the East Coast Mainline).
@@christophervalkoinen6358 Thanks for the info. That doesn't sound great...
When you're taking the train in France and the pa announces:
"The train from Eiffel Tower to Baguette has been delayed at Poincaré and will arrive in Oolalah with a 10 minute delay."
Then you're not in France, you're in a Numberphile video.
So glad the Singapore incident got a mention...I loved reading about that investigation when it happened. PS: Love the animations!
I need more about that
It is also during this early era that the concept of time zones was invented. Local sun time was becoming a big timing issue especially on faster East-West trains, like the ones in North America or in Central/Eastern Europe.
This was always a curious thing to me. Ships had been using clocks for east-west position for a fair time prior to time zones. The ships simply used a single fixed location to set the ship's chronometer such as the Greenwich observatory (Equivalent to using modern TAI or UTC) and compared it to the local sundial time to get longitude. If the chronometer says noon and the sundial is at 0900 then the longitude is 45degrees west. (15 degrees per hour, so 1 time minute = 15 angle minutes aka 15 nautical miles.) Not a literal sundial in most cases, they used a sextant to take finer angle measures and did some navigation math for more accuracy.
Aviation still uses UTC for all scheduling, conversion to local time zones is only used for display to passengers.
@@mytech6779 The problem wasn’t for the trains themselves rather than the coordination with local stations.
I was sure the video was going to be about time zones when I saw the title.
@@mytech6779 When every station declares "noon" the local solar zenith, and you are trying to work out transcontinental schedules, you *must* simplify the problem. That's what time zones do, instead of accuracy to the second in local time across the track. You accept a one-hour band that all use the same time standard even if they're on opposite edges of that band.
Now for the important question.
Is it better to live on the east edge of the zone, the west edge, or in the middle?
@@EdwinWiles Depends in if you are an early bird or a night owl.
1:22 I can't tell you how hard I laughed at this. Hannah's voiceover combined with the animation… Something about it really tickled me 😂
It amazes me that several major accidents had to take place in order for people to realize it was important that trains didn't run in opposite directions towards each other.
Apropos that oft-quoted story of Brunel saying he would pile on more speed; I reckon he was applying the well known idea that a silly question deserves a silly answer.
If _The Intemperate Engineer_ ( a biography of Brunel, mostly quotes from his letters) is right, "bigger faster heavier" was a typical Brunel answer to just about *any* question.
We also use these for signal timing on roads. They're called time space diagrams (TSD). You can visualize a corridor and how traffic stops, starts, and forms queues. If you've ever driven along a road and the signals turn green just as you get close, this is how the engineers did it.
Thinking back to my traffic engineering days, I could have sworn there was a more elegant word for it but I guess not! I think it’s time they got a more unique name
So now we know how they time the next set of traffic lights so they go red just as you reach them from the previous set - every time. Nice.
Timetable Planner here. I've never seen train graphs explained as well as this video does!
This train table graph contains space and time. It just looks like the Minowski Diagramm of Special Relativity!
My thoughts exactly, although the space and time axes are reversed from how they usually are.
Exactly what I thought of! Spacetime diagrams.
Friggin TIME ZONES were invented to handle this! Standardized time didn't exist until railway timetables!
Hannah Fry - you need your own channel. I will immediately subscribe.
Check out "Stand-up Maths".
I discovered these graphs a few years ago and became quite obsessed too. It's just so clever and simple. Still used today.
Wonderful to think about how they didn’t really care to ensure the giant fast-moving heavy metal machines wouldn’t run into each other until major accidents had already happened. Business truly never changes
I think what was said was a simplification. The Railway policemen had watches and would not let a following train go until a set time had passed. Perhaps 10 minutes. This was Ok if the train in front was faster than the following train and did not fail on the way. The problem was not really sorted by timetables but by the introduction of Block Signalling which is still used today.
Hannah Fry is great! More train content please!
In Construction Planning and Management similar techniques are used for planning and scheduling projects that a linear in nature, think bridges, highways etc. in that version they are referred to as linear schedules or line-balance schedules. Often used in conjunction with the typical CPM schedules it allows one to “see” the progress planned over time mapped to the physical location and identify constraints that may have been missed or opportunities for dividing the work or adjusting sequences to optimize the plan.
And for the full timetable you then have to solve the issue of the availability of rolling stock and personnel to have a train at all, and the time personnel can work in a day (and have them end their shift somewhere near their home).
I’d think making sure that a person’s shift begins near where their previous shift ended (especially if they’re working consecutive days) is even more important, and obviously they need that place to be near suitable accommodations, preferably their own home.
Don't forget to put in your walking, taxi and PNBs as well.
A transit analyst in Toronto, Steve Munro, creates the same type of graph to analyze bus and streetcar bunching and delays, using historical vehicle location data. If the diagonal lines are evenly spaced, buses came at regular intervals. If there are several very close to each other, it shows a series of buses that are bunched together. If there are lots of very flat or horizontal lines, it shows that there was a delay that held up buses, and shows where along the route buses are most delayed and at what time of the day.
it's incredible that you can solve huge problems with simple graphs like this. what a brilliant idea.
I knew immediately what this was about. I lived at a railway station as a kid and my father worked at the office. So, often I was playing around the office. I remember this graph and I was actually able to use it, as a 10-year old kid.
"Lived at" as in visited your father frequently at work, or slept upstairs of the platforms?
@@otm646 Our flat was in the same building, above the offices.
WOW. So many of my favorite things in one video. Math! Trains. Train history. Time. My mind is overloaded by how much I loved this topic! There's this old children's series with this Ralph guy who works all these different railroad jobs in various books. As a kids series it's obviously unrealistic how many incidents happen to this one guy. But one thing I liked was a look at what working at a railroad was like ( roughly) in the early 1900s. In one book, they spent a lot of time talking about railroad scheduling, which involved a lot of knowledge or the rail layout, and route timings, and dealing with "specials" which were sort of out of normal operating schedules. Not a simple task I am guessing, and getting it wrong could involve loss of lives and property. I presume things got easier with electronic systems.
I have been working for the Dutch railways for 17 years and we still use these graphs today. Given they are on a PC screen but it is still used. I started at the signaling department in Amsterdam, went on the train for 11 years and due to a work related injury today started "temporarily, but we'll see about that' on the "Production" department at HQ where we have all infrastructure in the system and design and simulate future timetables to see what is possible and not.
String graph(or diagram). Because ... strings+pins (of varying colours to denote different classes of trains), being adjustable without needing constant erasing and re-erasing, were used to _design_ the timetable and set schedules using a large board mounted on a wall. Once it all fit nicely the final version was drawn in reduced size on paper, printed, and circulated.
That was certainly the way it was done in Canada and the U.S. before the advent of computer programs that do just the same thing on a monitor screen. Hence the expression, well known to railroaders of the era, "stringing the time table".
Come to think of it, what you've got there is an analog computer, isn't it?
Hannah Fry is one of the most wonderful people on Earth.
The pure joy and excitement this video radiates!!!
I wished i had a math teacher like Hannah, when i was in school. I could listen to her all day talking about the wonders of math and science. Of course i got myself a signed copy of her latest book, so i can feel and learn like a child again.
I loved how many times you were able to get the new book in the video, including holding up the train tracks. Well done! Also, I just finished the book and loved it.
This was utterly fascinating! I love Prof. Fry's enthusiasm and way of explaining things.
I can't get enough of Hannah Fry! I love her so much!
A new Numberphile with Hannah Fry? Be still, my heart!!
Of course, this all pertains to European trains. Here in America I would imagine they probably still use train graphs, but they are drawn with wide paint-brushes so you can't tell what time they actually connect.
More likely spray cans - lines so broad and fuzzy they may as well be blurs.
More like most US railroads carry 99+% freight, and don't give Amtrak enough space in those diagrams to _stay_ on time.
I could listen to Hannah speak forever honestly
Although I’m a railway enthusiast, this type of train graph is literally beyond what my brain can comprehend. Now I do understand the part at 3:57, it’s pretty straightforward but when they start to add more than two trains to the timetable and they have to cross each other, that’s when my brain melts
This is fascinating stuff. Being a five day a week railway commuter between Essex and London I’ve often wondered how timetables were planned. Now I know and will sleep better because of this clip.
7:00
Additionally, the y-axis stations are spaced to scale (for example: 1”= 10 miles).
Double tracks & single tracks were different line types. Nodes added to indicate transition from single to double.
The birth of logistics.
Nice to see the old crew back.
I used to work in the Power Box at Wolverhampton station many years ago and even did my degree project on relieving congestion at Birmingham New Street based on mathematical modelling, multivariate analysis and using the A&D book I had access to, but in all of that and the rest of my life, I'd never heard of these graphs. Thank you so much for this video!!
I was saw a presentation a long time ago by the people doing the software for the Dutch train network. It was cool to see how visual it was. There were graphs to track the wagons across the network during the week (because trains are longer in rush hour and can only join in certain ways) and actual graphs with nodes to calculate the timetables, so they could model constraints based on what changes between routes they wanted to make easy. Super fascinating stuff!
just found hannah on youtube like 2 weeks ago! great to see her on here right after discovery!! she's awesome!
Watching her talk is absolute delight.
Absolutely loved this video! Great explanation and great animation :)
I work in the bus industry and we also sometimes use these plots for scheduling, I know them as "string plots". They can be useful on some routes to visualise where "bunching" is occurring (nothing worse than waiting for a late bus and then three show up together). And I've also found them very useful for explaining bus scheduling concepts with toy examples. They're also included in a number of software packages but in my experience they're not routinely used by schedulers. From the sounds of it they're more widely used in the rail industry. I wonder if this is due to them being more useful for rail scheduling or if it's due to the historical origins of these plots coming from within the rail industry.
I think it has to do with the problem when two trains meet being bigger. If two busses meet, and one needs to go faster, it can overtake the other easily on many roads. But for trains that would require changing some switches + an empty line on the way back, so you would have to account for it in the schedule.
@@mennoltvanalten7260 also train people take care not to have trains too close behind each other, you need a fair amount of buffer, whereas bus drivers can be relied upon themselves not to hit whatever’s in front of them.
ever since my college days studying data vis train graphs and bump charts have always been my favorite. some of the edward tufte books sitting on my shelf have beautiful examples of both that imo would look great blown up to poster size if a bit nerdy (very nerdy)
I got to come out and admit it, anything with Hannah Fry in I click it.
Love seeing Hannah here again
In the mid 1980s I was a consultant for a project to do computerized analysis like this for the Welland Canal -- the canal that allows ships to go around Niagra falls, going between the Great Lakes in the midwestern United States through Canada to the Atlantic Ocean. They hoped that with the help of a very expensive computer system, they could figure out a way to have slower ships wait for faster ships to go by, and increase the total volume of ships a little bit.
Did it work?
@@abdullaahimohamed2695 Yes. I believe (and this was many years ago, so it might have changed without my knowledge) that there are controllers in communication with people on each ship, and there are sensors that let the controllers know pretty precisely where the ships are. The computer system also accesses a database to tell the controllers how fast each ship is able to go. Then the controller can know in advance if a fast ship might get stuck behind a slow ship, and show much the benefit would be to having a slow ship pull over in a part of the canal that had extra room, and let one or more fast ships go by.
This is a great channel! I only caught sight of you folks in recent times after the 301 views video, but it’s now among my favourite channels.
In the US, train graphs (aka Time-Distance Graphs, TDG) are used on both mainline freight systems as well as urban/suburban transit systems, in most cases automatically generated via computer, showing the planned and actual train movements. They are used by control center dispatchers and controllers to monitor train movements, deal with incidents and delays, and then analyzed later by schedule planners to adjust schedules to maximize performance.
Love the videos with Dr. Hannah
Amazing video! That period of time in Singapore where the Circle (yellow) line kept breaking down was horrible. I know people who changed houses so that they won't have to take the Circle line. It affected the Green and Red line as well. People who took the Circle line were excused for being late to work, because of this issue. I've heard that there was a rising interest in working on a system that could technically certify when someone claimed they were stuck somewhere on the Circle line. Then after a while everything magically worked and the interest in that line of work also faded.
I remember seeing the 'train graph' in the old British Transport Films, really enjoyed watching this and learning more about these types of graphs
I used to ride a commuter train on the US east coast between Washington DC and Baltimore. It had to pull over every time a long distance passenger (Amtrak) or freight (CRT) train came by.
Awesome explanation! Hannah is so amazing!
Hannah Fry is a pleasure to watch and listen
Your voice is just so calming a bed time story with it would be so effective
I work as a bus scheduler (I make the duties for the vehicles and drivers, but may also suggest to chnge the timetables for efficiency reasons) and use these a lot - they make it easier to see frequencies and connections tha a printed timetable.
Even though buses don't have the collision problems trains have as they can overtake and pass each other (although we have a road where buses can't pass due to overhanging trees).
I love this! And not in the last place because routing and rostering currently is so incredibly complicated that this video almost breathes a bit of fresh air in that world. Sometimes making complex things tangible seems to be very effective. Would love to see a deep dive in the more complicated world of routing!
OMG Hannah Fry is back !!! Excellent !
I was beginning to think Hannah wasn't doing anymore of these, I'm glad she is my favorite math teacher
I have encountered so many things that I wish were taught in grade school when learning about sine waves, the square root of 3 and now this crazy graph that would have been awesome in algebra somewhere. Its amazing that it ran railways for over a century, and possibly will continue in a computer for centuries.
I really don't care about all of this information or the mathematics behind it, but it feels nice to watch someone who can be so genuinely excited about it.
I really laugh by heart with you naming the french stations!! As with good old friends, really appreciate it!
very interesting episode ! I am fan of train and math so this is a beautiful fusion of field. For information the station between paris rouen today are mantes Vernon Gaillon Val de Reuil and Oissel
There are also very detailed version for a small area with times to 30 trains per hour you need to consider precision timing at this level.
Human ingenuity never ceases to amaze me. Thanks for making this video. 🧐
It's always nice to see Hannah smile
8:12 "naughty train"... various things come to mind with that phrase
Great video! I first saw this visualization in one of Edward Tufte's books, but this is the first time I've really understood it.
I love how excited and just absolutely OUTWARDLY GIDDY Hannah Fry is 😂
If Dr. Fry only makes Numberphile appearances coinciding with the release of her books, I insist she become much more prolific an author.
What we see, as passengers, at the stations, tables with departure and arrival times, thats just the nodes where the lines are crossing.
And when you hear an announcement over the P.A. you know something knocked a kink into the lines somewhere.
A lot cooler than I was expecting. Great video!
Hi, Dr. Fry, I'm happy to see you on Numberphile again. Thanks.
Ooh! A numberphile video about 19th century train timetables!
This is my jam
It's always great having Dr Fry back.
As a French man having travelled on this very trainline when I was young, I can say that your station names are awesomely better than the real ones !