[Music] Alex: Hola, hola Brady.
Brady: Hola. Alex: Hola Brady, aqui el secencia, I think that's what sequence is in Spanish, de Bernardo Recamán who is a Colombian, a friend of mine actually. Hello Bernardo if you're watching He is a teacher in Colombia and he invented the sequence called the Recamán sequence which has got all the sequence people crazy because it's like really interesting and really fun. So it's really much easier to explain how this works on a number line. Okay, so just say this is naught and this is one two three Four, okay, five...eleven twelve thirteen and we go on okay So what we're gonna do we're gonna go in hops We're going to start at zero and the first hop is one and then the next hop is two Then the next hop is three. So each iteration you add one to the hop. Okay dead easy and the rule is "atrás siempre" always back But if you can't go back because the number has already been used then you got to go forward We start at zero, so then we add one, we hop one. One. I've got to go two we've got to go backwards Can't go backwards, so got to go forwards. Two Three. Can we go backwards? One, two, three? No We can't do that. Yeah, so we gotta go forwards one, two, three. We got to get to here Now four One two three four--yeah. We can, so we can go back there. What about five? Well can't go back, so we have to go five. One, two, three, four five. Yep, and go there five Six one two, three, four five six, no can't do that. So gotta go here. One, two, three, four five six Okay, you can see how that's gonna go. We're kind of gonna do these little weave in and out The question is, which is of interest and no one knows, is every number accounted for? So will this ultimately get in every number? And it's been tested up to some like some really high number But testing it's not the same as proving it. It's assumed that this will this will be the case. Yeah. I think it's cause you've got to go back. So you're going back and you're kind of clearing up for yourself. Now what's interesting about this is because it's very very simple. You're just jumping backwards and if you can't you go forward and you get this it's not chaotic, but it is... It's ordered for a bit. Then it is a little bit, not all over the place. Then it gets ordered again. And this is easier to see rather than just looking at numbers. Visually, and aurally. Either with your eyes or with your ears. Edmund Harris the mathematical artist who I work with a lot has done this gorgeous representation. And to me this looks like you know in the 1950s and 60s you would get these artists doing things that looked a bit like this. [Music] Kind of bit geometrical but also you got something a bit human in it also. Brady: It looks like it should be a wallpaper in Mad Men. Alex: Yeah, definitely. It totally could, you've got that mixture of something, which is just a circle, you couldn't get more simple than that, and it's got circles of different sizes and they all fit in together nicely But when you look in there's like a bit of a mess of how are they joining. There's a bit of chaos. It's standardly beautiful, but with a little bit of the unknown. [Music] In fact, if you saw here, this bit here, where it starts off with a Loch Ness monster, and then doubles back and goes on You've got that there. So the actual number line is the diagonal along here. Well what he's done, wherever you're joined, you actually have a proper semicircle. Whereas I've just done it with a sort of, I don't know what this is a kind of human sort of grasping at a circle. Where as when you when you do it And obviously, if it's a semicircle the radius is getting bigger. So the bit the bigger the size that the higher it's going to be. It's kind of getting bigger, but then it gets smaller again, but then he gets bigger and then it has getting smaller again Brady: Where does it end? It jumps out here, but it ends over here somewhere. Alex: Yeah no, so we've got the end there. That would be because there you can't go back and so you'd have to go on here And it's to keep it nice within the page looks like about a hundred that isn't it? I'll give a free book to whoever wants to tell me exactly where it ends. I'll give them a free book. Brady: They've got to buy the book to figure out where it ends. Alex: No, they don't even need to buy the book, they can make a screen grab of this But the thing is, you actually want to have two of these books because you want one to color in and like ruin and make your own. And you want to have one to have the perfect image Brady: By the way, I think Alex means he'll give a free book to the first person, not every person who does it. [laughter] Brady: Giving out 300 thousand books. Alex: Yeah. Alex: I first heard about the Recamán sequence from Neil Sloan. And Neil Sloan is the guy who invented, who founded, who runs the online encyclopedia of integer sequences. And there are about 300,000 different sequences. And I said, can you give me some example of sequences that you really liked? And one of the very first ones that he said was the Recamán sequence, and I said "Why is that?" He says because it has this interesting clash between order and chaos. And actually the best way to discover that is to listen to it, and on the online encyclopedia there is, on each sequence, well most of the sequences, all the ones that I've looked at There is a button, listen to, and if you want to listen to it you press it. So what it's gonna do, is that imagine on the number line Here we've got the numbers Imagine if this was a scale So that was the bottom of the scale and then you go up each one is a semitone going up bum-bum-bum-bum And you've got a sequence of I think it's six octaves, which is what most pianos might have. What you would do is that you would imagine that the first Sort of six octaves, so 48 numbers are These ones, going up. And then the next 48 go back to the bottom. So what you do is that you are playing the sequence as if it is a piece of music Every number the sequence comes exactly one after the next. It's not syncopated. It's like bang bang bang bang bang. One, two, three, four five six That's the world's most boring sequence. just goes 1,2,3,4,5 What you're gonna get is the scale [rising in pitch] doo doo doo doo doo doo When you get to the top: [falling in pitch] doo doo doo It goes grgrggrhrgh. I go at the bottom again, then carry on going up, really boring. [notes with steady increase in pitch] [sharp fall then steady increase again] If you're gonna do the sequence which is one-two, one-two, one-two, that's gonna be duh-nuh duh-nuh duh-nuh [rapid oscillation between two pitches] This interesting sequence, and how it sounds, and then we're gonna think, well why is that interesting? And I have it prepared here Brady: So here we go we're gonna listen-
Alex: Just downloaded it. [Rapid, somewhat chaotic notes with quick bursts of short-lived patterns] Alex: It sounds like someone is playing it [strange notes continue] And, you've got lots of things going on. You can see at the same time it's going up, but then it's going down. Then there's a bit that feels kind of like order, we know where it's going. And all of a sudden, just out of the blue and it's all a bit discordant. [rapid switching between pitches] I don't know, it's going up, it's going down and it's you know, it gives you the the chills. Brady: What does it remind you of? Alex: It reminds me of a horror movie or something that you know, maybe someone's going into a room. They gotta find something and you're supposed to be feeling nervous and tense and you'd be listening to that. There is something that feels there's a human hand behind it, but there is no human hand. We've got this incredibly simple rule, which is just back, but if you can't go back go forward And also the obviously the distances between the numbers are getting bigger and bigger and bigger. [notes seem to fall into flourishing pattern] There's a little twiddle there, where did that come from? Brady: Now if you're anything like me This video is gonna have sparked even more questions in your mind. And if that's the kind of person you are Maybe you should check out today's sponsor: Brilliant. Brilliant is a site full of quizzes and puzzles and courses, all sorts of things that are going to make you smarter, change the way you think, change the way you attack problems. And everything on brilliant is guided by eight principles. You can see them here and there at number eight last but definitely not least: Sparks questions It says the culmination of a great education isn't knowing all the answers, it's knowing what to ask. And I don't think there's anything truer than that. If you'd like to go and check out Brilliant, I really recommend that I especially like their kind of puzzle of the week. What have we got this week? There's basic intermediate and advanced. The basic one this week's really fun. It involves this rotating rod and light being flashed on it I recommend going and having a look. Now if you go to brilliant dot org slash numberphile, you can sign up with Brilliant and get 20% off a Premium membership. That means you get access to all the good stuff, 20% off. But just a reminder, you can go to Brilliant and look at lots of stuff for free including the problem of the week. So definitely go and check them out, and then I'd recommend going for the Premium Membership, so you get all the goodies. Brilliant dot org slash numberphile, our thanks to them for supporting this video.
