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\deflang2057\pard\qc\plain\f4\fs24\ul Frontiers\plain\f4\fs24
\par with \plain\f4\fs24\b Peter Evans\plain\f4\fs24
\par \pard\plain\f4\fs20 (Debussy plays)
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : A mathematician is having a dream,she's in a garden of sunflowers,the music of \plain\f4\fs20\cf1\ul Debussy \plain\f4\fs20 is in the air.So too is the scent of pineapples.Through some pine trees looms the unmistakable form of the parthenon of Athens. Round her feet rabbits are doing,well,what rabbits are renowned for doing.Our mathematician wakes up and immediately on her lips is name:\plain\f4\fs20\cf1\ul Fibonacci\plain\f4\fs20 .
\par So who's Fibonacci and why should a mathematician link together Debussy,rabbits the parthenon and all the rest of it? As well as answering those questions in this program,I'll be bringing up to date some mathematics of the 13th century,with an extraordinary recent discovery about the mysterious properties of a sequence of numbers.
\par But first a word of reassurance,if you're saying to yourself that your not mathematical,that you don't really understand what numbers, theorems, formulae, equations and proportions are all about.The fact is you do says Dean of Science at St Mary's College in Miraga,California,Keith Devlin.Try this little problem he set me.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : Well I'm going to ask you five simple questions,four that are in one block and then one that's going to be separate,okay? And you have to do this very quickly,in your head you don't have to read the answer out to me. Subtract 1 from 1,now do 4 - 1,now do 8 -7,now do 15 - 12.Okay now,think of a number between 12 and 5,got it?
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Yeah,yeah.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : You've got the number 7.Almost everybody listening in has the number 7. [Not everyone will because it is a probability that you will get this.It's likelihood is exploited by mind readers such as Uri Geller,I got 7 myself-LB]
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Why,why have I,come on?!
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : What I did was I set it up so that your ........you have a built in number sense,you have a built in pattern recogniser that works with numbers.In doing those subtractions of a smaller number from a larger number I got you to think about subtraction,then I said a number between 12 and 5,if you subtract 5 from 12 you get 7.You were subconsciously subtracting and thinking "ah 7 must be the answer".You didn't know you were doing that .I simply set it up so that your brain got involved in a pattern.So it's almost like been.....just as if I start to tap a beat out on a drum, you'd begin to resonate with that beat,if I start to tap out a mathematical pattern in your mind,your mind will begin to resonate with that pattern.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : So we're all innately mathematical,says Keith Devlin,we have a built in ability to latch on to number patterns,and that takes us back to the mathematicians dream,which links together rabbits and music and buildings to someone called Fibonacci.He was in fact a fellow mathematician.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : Fibonacci,his real name was Leonardo of Pisa.In many books you'll read that he called himself Fibonacci.Fibonacci would come from the latin phrase which means son of Bonacci.Modern historians of mathematics actually believe that the name Fibonacci was ascribed to him by an early 19th century writer who was writing about the history of mathematics.Be that as it may,Fibonacci is certainly one of the most significant figures in the history of mathematics.At the beginning of the 13th century he published a book called "Liba Abacci",which translates as "the book of the abacus",and it was that book that introduced to the western world the Hindu/Arabic number system that we now take for granted and that we use for our arithmetic.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : So without Fibonacci we'd be using Xs and Vs and III,probably?
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : We'd be still using the Xs and Vs that the Romans used,and as anyone would find out if they tried to do arithmetic with Roman numerals,arithmetic may be hard with Arabic numerals,but with Roman numerals by golly it's almost impossible!
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Good for Fibonacci then.But what about those lively rabbits.Well he wrote about them in his famous "Liba Abacci" - book of the abacus.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : Like all maths books,even a math book written in 1202 had some exercises for the reader,and one exercise that he gave,so that people could practise using these very efficient Arabic numerals was the following,he said "Imagine you've got a garden and someone puts a pair of baby rabbits in the garden.Now baby rabbits can mature and start reproducing every second month,and so these rabbits are in the garden from the second month onwards they start to reproduce,their children then mature and reproduce and so on.The question he asked was "how many rabbits will you have in the garden after one year?".Well if you think about it for a while,you'll see that you get a sequence of numbers coming out of it,and that after one month there's going to be one pair of rabbits,the next month it's the same pair of rabbits that are now becoming mature,then they have children so you have two pairs of rabbits,the next month you'll have three pairs,then 5,then 8,13,21,34,it's beginning to get a little bit overpopulated as populations tend to do.The rule that's generating these numbers is that each time you've got two numbers,you generate the next one by adding the last two.So 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, and so on and so on and so on.General rule,just keep adding your last two numbers that gives you your new number.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Enter the celebrated Fibonacci sequence.A string of numbers where each is the sum of the preceding two numbers.In the 13th century,Fibonacci used the reproductive capacity of rabbits to show it in action.But as Professor of Mathematics at Warwick University,Ian Stewart explains,this numerical sequence pops up all over the place.
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\par \plain\f4\fs20\b Ian Stewart\plain\f4\fs20 : The best applications of Fibonacci numbers,the one's I like are the ones where you go out,and you find yourself a plant and you find a flower and you count it's petals,and more often than not you get one of these Fibonacci numbers.If you look at pine cones and start counting the way the little bits and pieces of the cone are arranged.They're arranged in spirals,and you get Fibonacci numbers in the spirals.It's as if the plant kingdom is obsessed with these numbers.The pattern in sunflower [Ref: R.Penrose "Shadows of the Mind" p362;I.Stewart] seeds [And thus Red Indian dream catchers - LB],the pattern in pineapples,just the way that certain leaves will branch off from a stem,you find the Fibonacci numbers,or if not them,some mathematical relative which is very,very similar.The reason in plants is now fairly well known,and it's been a long story,it's been about 300 years to sort it all out,but the current version is it's a natural and inevitable consequence of the way the plant grows.When the tip of the plant,when it's a small shoot,just coming out of the ground,in there are little bits and pieces that are going to turn into the petals and other such things,and these pop into existence near the tip,and then they kind of migrate outwards,and they have to pack together,basically because if there was a gap then one of them would just kind of expand a bit and fill the gap.If your popping a large number of rather soft balloons into existence and they can pump themselves up to fill the available space,well obviously they are going to pack together,there won't be a big gap,because one of the balloons will expand its way into the gap,and this process,for reasons that would certainly take a little while to explain in detail,leads to a very specific kind of geometry.It so happens that in the plants,it's all growing from the centre and spreading out,it's a pattern of spirals,and Fibonacci numbers are intimately associated with every detail of the way those spirals work.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : So,mysteriously you might think,a man made sequence of numbers,is found to be identical to the way nature packs seeds and petals in plants. [This is not mysterious if you adopt the Platonic view that maths already existed in nature and man discovered it-LB] But that's not the end of it. If you take any number in a Fibonacci sequence and divide it by the one before,you get pretty much the same number,the relationship is constant.Keith Devlin again.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : That constant,surprise,surprise,surprise,and by golly it is a surprise is the classical \plain\f4\fs20\cf1\b\ul [Maths 4] Golden Ratio\plain\f4\fs20\cf1\ul \plain\f4\fs20 that the ancient Greeks had thought was very important in designing their buildings.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Which is what?
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : The \plain\f4\fs20\cf1\ul [Media 1] Golden Ratio \plain\f4\fs20 is approximately equal to \plain\f4\fs20\b 1.61803\plain\f4\fs20 ,like most mathematical constants, it has an infinite number of decimal places,it's what we call an \plain\f4\fs20\b irrational number\plain\f4\fs20 ,but back in 350BC or there abouts,the Greeks thought that this ratio was the perfect proportion. If you wanted to build a building or a stadium,the length divided by the width ought to be 1.61.If you built your buildings that way,they believed,human beings would find that the most pleasing.So for example,the parthenon,the front end of the parthenon,if you measure the width and the height and divide one by the other,you'll get the answer 1.618.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : The famous Golden Ratio or section then is nothing other than the proportion contained in a Fibonacci sequence.The "Divine" proportion as it was often called.Centuries before \plain\f4\fs20\cf1\ul [Maths 4] Fibonacci\plain\f4\fs20 ,the Greeks were exploiting it for their buildings.Later painters too were ringing the changes on it,to lend balance and harmony to their pictures.While in the 17th century a great violin maker found it\plain\f4\fs20\b useful\plain\f4\fs20 according to mathematician and Fibonacci officionado Ron Knot.
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\par \plain\f4\fs20\b Ron Knot\plain\f4\fs20 : Yes,yes I was surprised to see a reference in the New Oxford Companion to Music in volume 2,under the reference for the violin,there's a sort of picture there of Stradivari of the design that he had written,put down for his violins,and it seems to be using the Golden Section to place the "f" holes on the violin.Now we know that Stradivarius violins have the most exquisite sound and we're not quite sure why,whether it was the resins he used on them,whether it was the thicknesses of the wood or how the wood was aged or the shapes of it,we're not quite sure.But it's interesting that Stradivari did have a picture where there seemed to be Golden Sections on his diagram and it produces a very pleasant sound,so I'll leave that open to listeners to make up their own minds really.
\par ( Stradivarius violin plays)
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : The relationships and proportions inherent in a Fibonacci sequence of numbers haven't just shaped musical instruments.They've influenced musical composition too.Debussy's evocative study of reflections on water "\plain\f4\fs20\cf1\ul Reflects d'en Leau\plain\f4\fs20 " bears,for concert pianist \plain\f4\fs20\cf1\ul Roy Howard \plain\f4\fs20 the indelible mark of the famous sequence.
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\par \plain\f4\fs20\b Roy Howard\plain\f4\fs20 : Now the interesting structure of this piece is the way that its impressionistic and very imaginative and decorative surface disguises a very classical structure underneath.The piece is really built in sort of paragraphs,and each of these starts with quite a classical key,that you can easily identify in classical terms.The piece starts basically in this key (Piano chord sounds),after a while it moves away and you can't tell what key it's in until it it comes back again to this key (Same chord sounds).Now it takes 34 bars to get from the beginning of the piece to the return of that key.It then goes for another 21 bars until it moves to a new key (New chord sounds) (Debussy plays),and as it dies down that was the main climax of the piece.Here begins the next tonal definition it's getting back to the home key.It's very clear on the page,you can see the the key changes,it's written that way.So you have these paragraphs defined by this sequence which is being compressed by Golden Section each time (Five chords play).
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Debussy's "Reflects d'en Leau" ,where distinct mathematical patterns underlie apparently free flowing music.Now exploring patterns is what much of mathematics is all about,and that says psychologist Susan Blackmore makes it a \plain\f4\fs20\b quintessentially human activity\plain\f4\fs20 .
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\par \plain\f4\fs20\b Susan Blackmore \plain\f4\fs20 : The way we've evolved is to have brains that look for patterns in everything,that has helped us survive in the past,and looking for patters in numbers is no different.We're always looking to extract meaning,and now in a row of numbers,you can see here are two the same,here are two that add up,and the more you learn about simple arithmetic and maths and so on,\plain\f4\fs20\b the more complicated patterns you can see\plain\f4\fs20 .It's just a natural by -product of meaning seeking brains which is what we have.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : So Susan,given that our brains are so good at detecting patterns is there a tendency,a danger perhaps of seeing patterns where there aren't any?
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\par \plain\f4\fs20\b Susan Blackmore\plain\f4\fs20 : Whenever we're looking for meaning in things,there are always two possible dangers.One you\plain\f4\fs20\b fail to see meaning that's really there\plain\f4\fs20 ,and the other that you \plain\f4\fs20\b see meaning that isn't really there\plain\f4\fs20 [Mystics do both of these -LB],and we know that when people look at random numbers,which don't have any pattern in,they still will find patterns in there,and try to make something of them.That's what our brains do.But if you go too far in this direction, you can get yourself seriously into a muddle.Sometimes schizophrenics get terribly carried away with \plain\f4\fs20\b seeing the same number on a car number plate as they had on their ticket at the cinema last night and think this is meaning in the universe\plain\f4\fs20 [They don' have to be schizo,just innumerate or given to belief in synchronicity-LB],and so on, and this\plain\f4\fs20\b is a mistake\plain\f4\fs20 ,it's a natural human mistake,but taken to extremes it can be \plain\f4\fs20\b deeply problematic\plain\f4\fs20 .If you like, it's the price we pay for having the clever brains we have.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Now Susan Blackmore says that we can sometimes make the mistake of looking for patterns in randomness,where by definition you'd think there isn't any pattern to be found.But that's exactly what some mathematicians have been doing for over 40 years with an update randomised version of the Fibonacci sequence. [That's not the same thing Peter.Randomising an algorithms potential to do one thing or another is not truly random,and therefore could have a pattern-LB] Keith Devlin again.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : One thing that people often remark about the Fibonacci sequence is its very artificial.If you put rabbits into a field then they won't really grow 1,1,2,3,5,8 etc.,you'll have some dying,some rabbits won't love each other and they won't have children and you know,all sorts of things go on.Real life,unlike mathematical life is random [Or Chaotic -LB].So the Fibonacci sequence is somewhat idealised.The question that people asked in the late 1950s,or at least the question that mathematicians asked in the late 1950s,was " suppose you take the Fibonacci idea of a regular growth sequence,a regular growth process,and you allow for random events,you allow for \plain\f4\fs20\b chance\plain\f4\fs20 happenings,can you still a nice mathematical theory?".You might on the face of it think the answer was no,but in 1960, between 1960 and 1963 a couple of mathematicians called Hilel Faustenberg,who's now at the Hebrew University in Jerusalem,and Harry Keston who's at Cornell University in New York,they proved some....they developed some remarkable mathematics that shows that under certain general conditions,if you have a growth process,like the Fibonacci rabbits,but allowing for random events,you can still get ordered behaviour coming out of it.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : What would be a random event in a sequence of numbers then? That you wouldn't add but subtract, something like that ?
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : Okay,here's a simple example,and this will relate to something that happened very recently in the world of mathematics.Suppose you start to generate a Fibonacci sequence,but each time instead of just adding,you first toss a coin.If you get a heads then you add the last two numbers to give you the next number.If your coin comes up tails,you subtract.So now you've got a whole randomised version of Fibonacci,your allowing the sequence to grow and to decrease.If you do it that way,then depending on the way the coin tosses turn out,your numbers may get bigger and bigger,they can become negative,they can jump around.You can get very wild behaviour.Incidentally,if you remember\plain\f4\fs20\cf1\ul Tom Stoppard\plain\f4\fs20 's play "Rosencrantz and Gildenstern are dead",right at the beginning,Rosencrantz is tossing a coin, and he keeps getting heads every time.So if Rosencrantz played this game,he'd get the original Fibonacci sequence.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.(Flips coin) Heads.(Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : There is of course an art to building up suspense.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : Though it can be done by luck alone.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : If luck's the word I'm after.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : 76 love.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : Imagine.I take a coin out of my purse,and spin it in the air,it falls on the turf,tails I win....
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : ....I lose.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : 77.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : Times on the trot.Odd business.Imagine the odds.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : 50-50.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : 78 love.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : A weaker man might be moved to reexamine his faith,if in nothing else,at least in the law of probability.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : And the laws of nature generally,though not.....
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : (Flips coin) Heads.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : ....the law of gravity.
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : That sort of thing,as mathematicians say, "only happens with probability zero",which means essentially never.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Nick Trevethen,Professor of Numerical Analysis in the computing laboratory at Oxford University.
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : More often the series grows at a predictable rate that's smaller than the Golden Ratio.There are theorems,that's what mathematicians work with ;theorems,from the early 60s,proving that it must grow at a well-defined rate.But nobody knew that rate,and nobody had even really looked for it,partly because people must have assumed that you couldn't figure that sort of a thing out.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Why couldn't you figure it out?
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : Well those are philosophical questions aren't they? (Peter laughs) Certain physicists believe that there are some things that it isn't right to ask because they're almost impossible to know! You can predict what water will behave like near it's freezing point but you can't predict at exactly what temperature it will freeze,and some people might have thought that this was analogous to that.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Some people,but not a young Phd student,Divaka Viswaneth,who set himself the task of finding order in a random Fibonacci sequence,of finding a new number in an apparently formless string.[The mere fact that it has the Fibonacci sequence at its heart means it isn't formless-LB]
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : We are people who use computers.In this field of numerical analysis,you compute, and I asked him to find this number on a computer,and we worked and worked,he worked and worked,and eventually,months later he realised he could throw away the computer and find an exact solution.That amazed me.I had no expectation he'd be able to do that.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : This will be very difficult I think for you and indeed for our listeners,but can you tell me what it is that he did? I mean can you give us any insight into the kind of mathematics that he was using?
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : Well I'd love to try.His key ideas was to study what people call "trees" which are like trees that you and I know. [You might say he was seeing the wood for the trees ! -LB] Imagine a tree with a trunk and from that trunk come several branches,and from each branch comes a few limbs and on and on forever,so that the number of twigs and leaves is,as it were,infinite.Now Divaka showed that you could study this random Fibonacci sequence by measuring something on every limb of every branch on this infinite tree,and he found an exact formula for a certain quantity on each limb of the tree. [Sceptics might think it impossible to measure an infinite tree.Such such measurements involve the idea of a "limit" where a formula may head towards a "limiting" value.That value is a ceiling beyond which infinite sums cannot breach in principle,but towards which the formula is heading.This limit can be found without doing an infinite number of calculations-LB]
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : And the number that emerges from Divaka's formula is a new constant that had eluded mathematicians for 40 years.Words like "staggering" and "astonishing" surround Divaka's discovery,so what does it feel like to find yourself in the mathematical history books,while still in your early 20s? Divaka,now at the Mathematical Sciences Research Institute in Berkeley, California told me first what the new quasi - \plain\f4\fs20\b magic\plain\f4\fs20 number is.
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\par \plain\f4\fs20\b Divaka Viswaneth\plain\f4\fs20 : The number for the random Fibonacci sequence is where you do both additions and subtractions is 1.13198824.So when I saw this thing in the computer,it was quite clear already that this single number which gives you the rate of increase of terms of the sequence,was by itself an interesting quantity.But the analogue for that number with the Fibonacci sequence is the Golden Ratio,so if you take the original Fibonacci sequence,where you do just additions and not both additions and subtractions,and you look at the terms of that sequence,they increase regularly too,and the number that gives the rate of increase there,is very famous,it's called the Golden Ratio,and it was known to the Greeks.So the number here,it's an analogue of that.So just because of this analogy with another problem which has been known to mathematicians for a lot of years,it was sort of clear that the number itself would be an interesting thing.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : Now although Fibonacci didn't realise it,his famous constant was to be used over and over again in art,music,architecture,even biological science.\plain\f4\fs20\b It's turned out to be practical. Practicality though,isn't what motivates Divaka.\plain\f4\fs20
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\par \plain\f4\fs20\b Divaka Viswaneth\plain\f4\fs20 :\plain\f4\fs20\b \plain\f4\fs20 If you sort of look at what motivates mathematicians,it's been the same thing over several centuries.For example when Fibonacci first looked at his sequence,\plain\f4\fs20\b he did not really realise it had any application\plain\f4\fs20 . He just posed this problem,he did not even solve it completely.He just noticed that if you started with 1 and 1and you added every two terms to get the new term,then you sort of get an interesting pattern,that could be related to a few other things like something to do with the rabbit population.So \plain\f4\fs20\b the basic thing that motivated him was just curiosity\plain\f4\fs20 , and it was exactly the same when I looked at this problem too.So there was this random phenomenon,you just start with a set of numbers,and you randomly either add or subtract them,and miraculously enough,there is this single real number which describes very well what happens to your sequence when you just combine your additions and subtractions in a random way.So the fact that there \plain\f4\fs20\i is\plain\f4\fs20 such a real number,which can capture a phenomenon,which seems to have no order in it,was itself quite fascinating.So it was sort of the \plain\f4\fs20\b mystery\plain\f4\fs20 of the whole thing,that there is this one number which tells you,there is a pattern in this kind of a random sequence,and\plain\f4\fs20\b I just wanted to know\plain\f4\fs20 what that number was.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : So this newly found number,1.13198824,\plain\f4\fs20\b is it simply a mathematical plaything\plain\f4\fs20 ? Or \plain\f4\fs20\b could it,like the Golden Ratio,have any use in the real world\plain\f4\fs20 ? Divaka Viswaneth's Phd supervisor Nick Trevethen again.
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : The Golden Ratio or the Fibonacci sequence is important as a prototype of all kinds of other processes,and I like to think that this is a nice example that may serve as a prototype for random processes.
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\par \plain\f4\fs20\b Peter Evans\plain\f4\fs20 : When you say random processes,d'you mean random processes,apparently random processes that might occur in nature,it will enable us to get a handle on such things as.I don't know, \plain\f4\fs20\b Chaotic\plain\f4\fs20 complex systems like the weather for example,or the behaviour of molecules in a chemical plant,that sort of thing?
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\par \plain\f4\fs20\b Nick Trevethen\plain\f4\fs20 : That is exactly the kind of process I have in mind.Things involving mixing. Mathematicians call this "ergotic" behaviour.I must make it clear,I don't think Viswaneth's solution will directly solve any big problems of fluid mechanics or mixing,but it may serve as a prototypical example that will help people further refine their theories in those areas.
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\par \plain\f4\fs20\b Keith Devlin\plain\f4\fs20 : If we look at history,we find time and time again that these esoteric results,within 10, 15, 20 maybe 100 years turn out to be extremely useful,and let me just tell you one story to illustrate this. One of the most famous English mathematicians of all time was G.H.Hardy of Cambridge University.He worked on....in number theory,he dealt with these kinds of Fibonacci type sequences. That's exactly the kind of mathematics he did.Cambridge in the era when he was doing his work,at the early part of this century,and to some extent even today at Cambridge to be quite frank,\plain\f4\fs20\b one was proud of the fact that what one was doing was really not relevant to the real world.It was meant to be pure art,a high form of pure art\plain\f4\fs20 ,and in a book he wrote Hardy said,with clear pride he said, "Nothing that I have done in my life\plain\f4\fs20\b will ever be of any use\plain\f4\fs20 to mankind".Today,Hardy's work is the very heart of the cryptographic techniques we use when we send messages across on the world wide web,when we send military signals from one installation to another.Hardy's work is now central to civil defence,military defence,commerce,banking,the way we live our world.Hardy didn't know that was going to happen,but it did happen.Who knows what's going to happen with Viswaneth's work in 50 years from now? I'd.....if I had to make a bet,I'd say it would find an application that may well be a very significant application to society.\plain\f4\fs20\b Mathematics just does that\plain\f4\fs20 !
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\par \plain\f4\fs20\b Divaka Viswaneth\plain\f4\fs20 : It's very fascinating by itself,but I hope it finds an application in the future,and I'd be delighted if it finds an application in some part of science in the future.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : Excuse me,I think you're standing on my friends coin.
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\par \plain\f4\fs20\b Man\plain\f4\fs20 : So I am.
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : Come on let's go.
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : I say! That was lucky!
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\par \plain\f4\fs20\b Gildenstern\plain\f4\fs20 : What?
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\par \plain\f4\fs20\b Rosencrantz\plain\f4\fs20 : That coin,it came down tails!
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