Hannah's latest book (with Adam Rutherford) is the Complete Guide to Absolutely Everything (Abridged): amzn.to/30Mr6UE More Hannah on Numberphile: bit.ly/hannah_vids

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.

Hannah recently recovered from cancer treatment. Glad to see she is doing well.

Great to see Dr. Fry again here on Numberphile--her enthusiasm was one of the things that originally attracted me to this channel.

I love the way Prof. Fry lights up enthusiastically about the mention of gruesome train accidents.

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.

As a french man, I'm personnally offended that you didn't include Croissant on these train tracks.

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.

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. 😀

Hannah is adorable, it’s always fun seeing someone getting this excited about something they’re passionate about

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 !

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!

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.

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.

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.

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.

Tufte would love this. Amazing how some seemingly impossible problems become solvable with the right visualization.

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

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 :(

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.

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

In dunno why but dr. Fry makes Math so interesting and fun to listen too!

Dr. Fry's voice & accent is done of the best ASMR I've ever heard! 😍 I'm melting!

I'm French and I just wanted to say that I love those train station names! Great video!

So glad the Singapore incident got a mention...I loved reading about that investigation when it happened. PS: Love the animations!

2 mins in, and I'm loving the gung ho attitude of these ancient people

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 😂

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.

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!

This train table graph contains space and time. It just looks like the Minowski Diagramm of Special Relativity!

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).

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.

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.

I discovered these graphs a few years ago and became quite obsessed too. It's just so clever and simple. Still used today.

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.

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.

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.

Friggin TIME ZONES were invented to handle this! Standardized time didn't exist until railway timetables!

Timetable Planner here. I've never seen train graphs explained as well as this video does!

Hannah Fry is great! More train content please!

Hannah Fry - you need your own channel. I will immediately subscribe.

it's incredible that you can solve huge problems with simple graphs like this. what a brilliant idea.

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.

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).

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.

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 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.

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)

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.

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.

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.

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.

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.

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.

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.

The pure joy and excitement this video radiates!!!

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.

Hannah Fry is one of the most wonderful people on Earth.

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 can't get enough of Hannah Fry! I love her so much!

this graph is used to this day to manage traffic at the kiel canal, they have it software supported now, but still the same principle

I could listen to Hannah speak forever honestly

I was beginning to think Hannah wasn't doing anymore of these, I'm glad she is my favorite math teacher

A new Numberphile with Hannah Fry? Be still, my heart!!

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.

Watching her talk is absolute delight.

just found hannah on youtube like 2 weeks ago! great to see her on here right after discovery!! she's awesome!

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.

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!

Absolutely loved this video! Great explanation and great animation :)

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

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.

Love seeing Hannah here again

I got to come out and admit it, anything with Hannah Fry in I click 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

Your voice is just so calming a bed time story with it would be so effective

I really laugh by heart with you naming the french stations!! As with good old friends, really appreciate it!

This was utterly fascinating! I love Prof. Fry's enthusiasm and way of explaining things.

Hannah Fry is a pleasure to watch and listen

Nice to see the old crew back.

I love how excited and just absolutely OUTWARDLY GIDDY Hannah Fry is 😂

OMG Hannah Fry is back !!! Excellent !

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 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.

Hi, Dr. Fry, I'm happy to see you on Numberphile again. Thanks.

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 !

Ooh! A numberphile video about 19th century train timetables! This is my jam

Coming in late, apart from the graphs they also needed a system for all the stations being on the same time so they had to synchronise clocks. This was done using the newly invented telegraph to sens a signal along a wire to all the stations so that the clocks could be adjusted.

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.

A lost opportunity to name the train stations after famous French mathematicians... some come to mind: - Laplace - Fermat - Pascal - Lagrange - Fourier ...

Awesome explanation! Hannah is so amazing!

This reminds me of the planning in the Netherlands story I was told that it was all originating from the Willemsspoorbrug which dictated all train traffic in the entire country because of it's frequency of usage by trains and marine traffic which was so heavily regulated. I used to be shown those graphs for the the massive amount of trains going over that single bridge and how they started drawing from there since it was always the most congested place. edit: before 1994

Love the videos with Dr. Hannah

Transport engineers used this for years and years. In Dutch it was called a "tijd-weg-diagram", a literal translation of that in English would be a 'time-road-diagram'

If Dr. Fry only makes Numberphile appearances coinciding with the release of her books, I insist she become much more prolific an author.

Human ingenuity never ceases to amaze me. Thanks for making this video. 🧐

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.

Good to see Hannah Fry on Numberphile.