Planing Sequences (Le Rabot) - Numberphile

Key Takeaways

so i want to tell you about a letter i got a little while ago that i only just got around to reading [Music] actually it was in 2003 that it arrived but i only just found it on the floor my study is very messy so the author is a contributor to the oais a former contributor i have how you think i have to say called claude lenomo he suggested two transformations of sequences his second transformation that introduced a word i didn't know in a french word called um lebeau this is a rebel a player carpenter's plane and his transformation is to take a sequence or a word a string of numbers and playing them down or about a planing the sequence what you plane down are the runs in a sequence if we have a sequence you might be tossing a coin and you might get heads heads tails tails tails heads tails you look at the runs here we've got a run of two heads so we get a two we write down the length of the runs three one one so the runs transformation of a string a sequence is the lengths of the runs that's the runs transformation what the the plane does it shortens all the runs by one by one turn so if we had a run of three things in the transform thing we've got a run of two things so we shorten all the runs that's the operation of rapote so you remember in one of the earlier number file videos you did you mentioned a sequence of my old friend sol galam and this is distinguished by the fact that it's a sequence of numbers it's the as the property that there it's an increasingly i should say non-decreasing sequence it keeps growing and it has the property that if you look at the runs length you get the same sequence all right i'll just do a couple of terms one two two how do we continue golem sequence the one means the first run has length one so we start off with a run of length one and then we've got a run of some other numbers and you always pick the smallest number so the next number has to be a two that two says the second run has length two so we've got two twos and now that two which is the third term says the third run has length two so so yeah so it continues three three and then four four four five five five this will probably make it clearer so this two refers to those two the third term is two and that says the third run of successive identical terms has length two well okay we've got our first run our second run our third run has to have length two so that means we've got two threes the fourth term and that says the next run has length three so there are going to be three fours that three says three fours this three says aha three five so there'll be four sixes there are gonna be four sixes very good and there are going to be four sevens from that four and that four says there are going to be four eights and that five says there are going to be five nines that's the defining property it's non-decreasing you always put down the smallest number you can and it must be its own runs transformation so that's column sequence now if we apply the rubber to it and we plane it down what are we going to do well we're going to cross off that one we've got two twos so we throw away one of them we've got two threes we throw away one of them we've got three fours we throw away one five we just shorten all the runs we apply the plane the rebel and what we get the new sequence is simply two three four four five five six six six seven seven seven and so on so this is the plain down version of golem sequence in claude's letter he gave the example of the golem sequence and how to rubate it so i thought it would be very interesting to apply the plain la rabo to the numbers to the ordinary numbers written in binary now remember the rule is when you plane it down if it's got a run of length one it goes away so zero well becomes zero so we get zero one goes away and we get zero one goes away zero goes away and we get zero one one becomes one in base 2 which is 1 in base 10. so our new sequence is what you get when you plane down the numbers so we get 0 0 0 this is the base 2 version this is the base 10 version so one note naught we lose the one and we shorten that zero and we get zero one zero one zero base two is zero base ten say here now finally we get something interesting we lose the zero and we lose a 1 so we get 1. here 1 1 1 we lose the 1 we got 1 1 which is 1 1 which is 3. so the sequence that we're going to get goes 0 0 0 1 0 0 1 three this one we lose that and that and we get zero this we lose the one and that and we get zero and ten we lose everything and eleven we lose that that and that and we get one twelve we lose a one and a zero and finally we get a two we're doing thirteen we lose that and that and that and we get a one because 12 is one one zero zero it becomes one zero which is two thirteen which is one one zero one we lose the one the zero and one of the ones and all we are left with is a one when we get to 14 we lose a zero and a one and we're left with one one which is three and one one one one which is fifteen we lose the one we get one one one and we get seven and when we get down to 16 we get zero again so that's and we keep going forever so this is the plane down version this has a very nice sound you know you can listen to any any of the 300 000 sequences in the oeis and i would like to play this sequence for you [Music] [Music] how have you turned into music we imagine that we have a grand piano that's the basic thing and we number the keys 0 1 2 3 up to 87 it's got 88 keys we take the number and we divide by 88 and keep the remainder we reduce it mod 88 so that gives us something we can play on the grand piano so if they're these numbers you can see they're all pretty small so it stays down here at the bottom of the keyboard for quite a long time until we get up to some very big numbers before we actually have to round them down why do you like that piece of music yeah why do i like it well it has that rhythm to it it goes da da da da da da and those those repetitions are because of the binary expansion and then every so often like at fifty pounds of two minus one we get a bigger number and before it we get some other big numbers so there's a rhythm so the sound the sounds reflect the structure of the binary numbers but processed is not just the straight binary numbers [Music] neil as you keep planing the binaries obviously we were we were staying pretty low though with those numbers do you start eventually getting some big numbers when you turn them into you know base 10 yeah certainly when we got up to say 2 to the 20 minus 1 playing off one we get 19 ones in a row so eventually we get all does every number appear here sure okay so there's an inverse operation which is extending all the runs by one bit and that works for any number what's that called if that's if that's what do you call the other version i just called it expanding claude didn't mention it in his letter but obviously you could do it you expand all the runs by one and then if you applied his operation you'd get back to where you started so you can get every number so a question is after you've planed down a number how big is it what's the average value after you plane it down so i looked at that and i asked some friends for help and they proved my conjecture is correct if n is k bits long in binary in other words if it's the k bit number so for instance here are all the three bit numbers there are actually four of them a three bit number beginning with one of course apart from zero every binary number begins with one so we could have one naught naught all the way up to one one one so there are um two to the k minus one three bit numbers there are two squared numbers there are four of them what's the average value well it happens to be we had four numbers um whose sum is four so the average value afterwards is one so what happens for a k bit number i conjectured that the average value after is exactly three halves to the power k minus two minus a half and the way i found that was i worked out the average you can see i just worked out the average for three bit numbers and i got one if you do that for the first 20 or 30 terms you get a sequence what you do when you get a sequence you look it up in the oais and it is sequence a 27649 which is the integer three halves to the k minus one minus a half so the answer to this question how big is what you get it's about three halves to the k so n was about two to the k and we shrunk it down to three halves to the k so that's the answer you might remember these pieces of pie we shared with number five supporters a while back well they were so popular we've now created these unique views of our beloved mandelbrot set each piece is individually numbered and is a one-off perspective a different location a different magnification if you'd like us to send you one find out more in the link we've included below i want to talk about approximating pi in the mandelbrot set maybe it's best to do a little bit of a reminder about what the mandelbrot set is so remember the mandelbrot set was this object that we have in the complex plane so here are the real numbers and here imaginary numbers and the mandelbrot set was a collection of complex numbers so i'll try and sort of draw a nice version of this here