Fibonacci Mystery - Numberphile

Key Takeaways

today I want to do a video response a response to one of our own number file videos uh because uh some time ago uh Brady made a video with our number file composer uh Alan Stewarts uh it was a video it was 40 minutes long this video which was a test to our loyalty I think uh but in that video if you made it through Allan uh described composing one of his pieces about the Fibonacci sequence it involved the Fibonacci sequence and he came up with a question uh as a challenge I guess to the number files I'm not quite good enough at the maths to be able to explain why that is maybe a number file will be able to explain why that is I'm going to answer his question today so what Allan was doing was uh he was trying to involve the Fibonacci sequence in a piece of music so first let's have a recap of the Fibonacci sequence very important sequence in mathematics it's quite easy it starts with one then it's one again and then you add the previous two values so you add these two and you get the next number so that's just 1 + 1 = 2 then you add these two values so that's 1 + 2 = 3 then you add the previous two that's 2 + 3 = 5 and you keep going in this way 987 6765 and so on okay so there's the Fibonacci sequence and what Allan was doing was to turn this into a piece of music he was dividing by seven he's going to divide all these Fibonacci numbers by seven now you can't divide exactly by seven most of the time H so he he looked at the remainder what you get left over when you divide by seven that's one that's two that's three that's five now eight when you divide eight by seven that's one with one left over that's what he wrote down then 13 / 7 is 1 with six left over 21 ID 7 is three exactly it has zero left over so you write zero and we're going to continue in that way we're going to do all the remainders two three and so on so he wrote out all the remainders and to make it a piece of music he turned these remainders into musical notes so it corresponded to notes uh but he didn't notice a pattern when he did it curiously he found the pattern repeated it actually repeated every 16 numbers so we go 1 1 2 3 5 1 60 66 54 261 0 1 1 2 3 and then the pattern repeated and he asked what is this about is this a thing and it is a thing it's actually called a Pano period Pano period is named after Leonardo Pisano and if you think you haven't heard of him before ah you're wrong because that's just another name for Fibonacci so this is a Pano period now Alan chose seven to do this because that helps him make his musical notes if he had picked any number though he would have got a period as well but the length of the period would have been different so if you divided by two if you divided by three if you divided by 9 or 700 he would have got a cyclic pattern uh if we divide by five as an example this should be quite easy divided by five this is when I'll make a mistake one one two left over 3id five that's three left over 5id 5 is 0 0 left over 8 divid by 5 is a one with three left over that would be a three four left over that has one left over that has zero left over and I think that's actually the full pattern uh the when you divide by five the period is 20 so this is the whole thing here at this point it will start to repeat again the zeros I want to point out are interesting the zeros are when you can divide by five exactly uh so there's a zero here there's a zero here there's a zero here as well oh look at look at this no look I've just noticed there's a mistake I said that look I said this had a remainder of five but look I don't need this that has a remainder of zero that just shows you're doing it all live doesn't I I am doing it now I noticed that that was a mistake I noticed there was a problem there because I knew that the period has only uh can only have one zero two zeros or four so that's another result in mathematics so because there was three I knew there was a problem there and I found it so that's another example of the Pisano period if you another thing I want to point out then it's actually every fifth number every fifth number there is a there is a proper bit of maths behind this we might as well do it there's another result about Fibonacci numbers it says a Fibonacci number let's call it the nth Fibonacci number exactly divides which I use a vertical line for another Fibonacci number which I'm going to call FM the result is if and only if n divides M so the little index at the bottom here if they divide then the Fibonacci numbers divide as well just to give you an example then we were doing if we look at the fifth Fibonacci number that is equal to five so uh so five divides Fibonacci numbers if and only if there's a little symbol there five divides the value of the index so every fifth number every fifth Fibonacci number is divisible by five that's what that shows let me just do one more example of that just to make the point if I took the next one look the sixth Fibonacci number is eight so eight will divide Fibonacci numbers exactly when uh six that's the index here divides M so every sixth number every sixth fibin Archy number is divisible by eight so this Pano period idea was first discovered by a mathematician called lrange in 1774 and he was actually looking at the patterns when you divide by 10 when you divide by 10 what you have left over is just the last digit let's do a quick version of that what colors do I need now I'll have to use black 5 8 13 / by 10 is 1 with three left over 21 divid 10 has one left over and what he discovered was the they had a they did have a pattern and the pattern had a period of 60 so the last digits of the Fibonacci sequence have a pattern of length 60 if you divide by 100 then you're actually looking at the last two digits of uh of the Fibonacci sequence that has a pattern of 300 length 300 if you look at the last three digits that's dividing by a thousand that has a pattern of length 1, 1500 the other easy way to do this is you don't have to write out the Fibonacci sequence and do each calculation for each number actually there's a property of these remainders that you can just add up the previous two remainders let's have a look so if we look at an example like uh here this is when I was dividing by seven four the remainder of four plus a remainder of two gives me the next remainder it's just like the Fibonacci sequence it gives me a remainder of six remainder of two plus a remainder of six it actually wraps back because we're dividing by seven when you go past seven it wraps back to zero so this will take me back to one then 6 + 1 will give me seven which wraps back to zero so I can actually do these sequences by simply adding up the remainders so using this idea when you add the previous two remainders gives you the next one actually kind of explains what's going on when I have add a one and a zero here next to each other then when I add them together I get a one then I add 0 plus one and I get another one so I've got two ones in a row one and one and then that's the beginning of the Fibonacci sequence and then it just carries on so whenever you have a zero next to a one you're going to end up back into the Fibonacci you're going to end up back to the start of the Fibonacci sequence and it has been shown that this will always always happen so that 01 is like a big trigger point yeah yeah that's going to send you back to the start again and that will happen in the other sequences that we had here as well look this is the dividing by five it has a period of 20 that's the end of it and there's the zero and the one again and that's going to send you back to the start so you're always going to end up back to the start this period is always going to repeat however there is no general formula for the length of the period so that's something we don't know yet at that two boxes of nine and a box of six there we go that makes 44 [Music]