2. The Golden Ratio & Fibonacci Numbers: Fact versus Fiction

Stanford University so well welcome back everyone the silent sheet is wandering around if you haven't yet signed up sometime within the next couple of days I was away at the end of last week I was in but Bailey University in Texas and didn't have time to do this but sometime I'll go through the list and see those of you who want to do this for credit and I'll send out an email and we'll sort of start some negotiations about what kind of work you've you interested in doing to do it on a credit basis and I want to pick up on on a theme that we just really touched upon last time when I was talking about the introduction of numbers to Western Europe and modern hindu-arabic arithmetic and in a mention that Leonardo of Pisa in his mammoth book Libra about she had this little throwaway problem about rabbits that give rise to this whole cottage industry about building up beliefs about the Fibonacci numbers and they a lot of interesting related number the golden ratio now unusual for for a course like this there's actually a quiz and I'm about to give you the quiz but it's going to be a self graded quiz and as we go through the lecture the first part of the lecture I'm going to tell you the standard story about the golden ratio and the Fibonacci numbers that you can read in books and websites as much as you want in fact if you go back to anything I wrote over about 15 years ago you'll find the same claims made in my books and articles if you go back onto the NPR archives and listen to when I talked about the golden ratio of many years ago you'll find I made certain claims and it's not altogether clear that all of those claims are true so as we go through the lecture the first part which is the standard story what I want you to do is just mark off what do you think each one is true or false as we get to it and then after I've gone through the standard story we're going to pause take our breath and then I'm going to go through and examine all of the variable leave some claims about the golden ratio on the Fibonacci numbers and there's a lot of them okay this is this is one of those places where mathematics some of it very interesting pure mathematics some interesting results in number theory to do with the Fibonacci sequence so we're going to have this place from mathematics real mathematics meets sort of popular culture and when that happens strange things can kind of care so this is the facts versus friction we're going to try and tease apart among these beliefs and claims about the golden ratio which ones are true and which ones are false and now we're also going to do a little bit of experimental work ourselves to just see if we can get some empirical evidence that would support whether something is true or false so the number right on the top is the first few decimal places of this irrational number the golden ratio already that's kind of interesting because a ratio is rational and this number is irrational so the golden ratio is actually not a ratio if numbers in the sense of whole numbers it's a ratio of lengths Oh a rational number is just one that's a quotient of two integers two whole numbers one divided by the other so any number which is obtained by dividing one whole number by another whole number is a rational number and until the time of the pythagorean's it was assumed that all numbers were rationals because you can take by taking big enough denominators you can make numbers as accurate as you want for measuring things and indeed the rational numbers are adequate certainly adequate for for measuring things in fact you usually don't need it in nominating more than about a hundred or so in order to measure things quite accurately but then there was a discovery made by one of the pythagorean's that if you take a right angle triangle whose sides adjacent to the part to the to the right angle a one unit so that the by Pythagoras theorem the hypotenuse is the square root of two you've got hypotenuse e squared equals the square root some of the squares of the other sides so you've got one squared plus one squared so the hypotenuse squared is two so the hypotenuse itself is root two and one of the pythagorean's showed that the square root of two cannot be expressed as a quotient of two integers in fact just today when my online Mook my online course with this course with 61 thousand students when the video that went out today actually contains a proof of that that ancient results so just by chance on today's course the students of sixty thousand students were able to see the proof that the square root of two is irrational okay so and the feet one of the features about irrational numbers is that their decimal expansion because they're not a quotient of two whole numbers the decimal expansion continues forever without settling into any repeating pattern like a third continue forever it's point three three three recurring but as an obvious pattern to it irrational numbers there's no pattern emerges to the expansion so if you continued that number on the top to generate this thing called the golden ratio which we'll look at then it will keep on different numbers will keep coming up and surprising you all the time now the sequence at the bottom is the Fibonacci sequence that we met last time and as we'll see today those two numbers are actually very closely related which tells you that there are this is more than just rabbits there's something else going on it's kind of interesting but as it and the symbol Phi is a symbol that relatively recently has been used to denote the the golden ratio okay so let me tell you the standard story and now someone got a copy of the quiz now because you're going to you're going to grade yourself I mean you're going to put true and false in as you go through the lecture and then after we've when we reach the pause before I go through and examine them if you filled in all the T's and FS then you can go through and grade yourself okay and then we'll see if anyone can get them all right okay okay so here's the standard story about the golden ratio it's a great story okay one of the ways of representing it or explaining it is to say and the claim is that there's an item as a rectangle whose aspect ratio the human cognitive system finds the most pleasing so there's an aesthetic thing that the golden ratio is the aspect of a rectangle which the human eye finds the most pleasing okay that's one of the claims made about it that rectangle and here is one of them sort of the golden rectangle the aspect we assure is known as the golden ratio sometimes called the divine proportion okay that goes back to Leonardo da Vinci's time I mean that these terminology goes back a ways so three different terms all talking about the same thing and here's one way you can get it now I mentioned that it's actually not it's an irrational number so it's not in fact the ratio of two whole numbers it is however the ratio of two lengths and what happened in to the ancient Greeks is there we know they're interested in geometry so they were thinking that pairs of numbers quotients of numbers could measure all lengths when they realize that wasn't the case essentially geometry and and arithmetic split apart and saw there were numbers rational numbers and you could do things with them and then there was lengths in fact when even even as recently as the 17th century when Newton invented calculus or Newton and libraries invented calculus it was actually applied to lengths not to numbers because it wasn't till the end of the 19th century that mathematicians finally fused together those two conceptions of numbers the ones that came from our cognitive apparatus dealing with measurements and the ones that dealt with counting and it was a rather difficult task to bring those together okay and it occupies most students the whole first year of university the ones that do mathematics took a go through the material it shows how those things come together so here was the here's one way you can get it you divide a line into two so that the ratio of the hole to the larger part equals the ratio of the larger part to the smaller so here's a picture there's a line and I'll choose my units so that the line has length one I'll split it up into a part that's X and the other part is one minus X then the ratio of the whole which is one to the larger part which is x equals the ratio of the larger part to the smaller part this this idea of doing this goes back at least to Euclid this was in Euclid's elements where you can take that e you can rewrite that equation that equation we've just seen you cross multiply well this is like one of these Tom Lehrer things if you're my age you cross multiply if you're younger you multiply the whole equation by x times 1 minus X we write that as x squared plus X minus 1 equals 0 just do whatever you do to turn it into a coin equation and then you solve it by either using the formula for the quadratic or you you use completing the square or whatever whatever your favorite way of solving the equation is and then we'll look at that in the second lecture as with last week there's really two lectures and that's going to be the pattern for most of this term okay you end up with this number zero point six one eight or three approximately and then the ratio which I'll start using this simplify is that is the ratio between the whole length and the x so the way they do so we now found the x we found the long part and so that's one point six one eight or three the the decimal part with Peaks okay so there's the this thing and it is irrational it keeps going on forever and it's irrational because of the square root of five just as the square root of two is irrational so is a square root of five well that's fine that looks like the kind of thing you'd expect to see in Euclid's elements but there's very little in Euclid's elements where a Google search will reach take about 4 million hits when you type in golden ratio so this is obviously a number with its up with a life of its own as well see it's related this number is related to the Fibonacci numbers and if you type in Fibonacci numbers they're far less common but you still get almost a million hits now there were very few numbers that get this kind of attention so these are clearly cultural numbers they mean something to a lot of people in fact it's amazing what they mean to a lot of people because if you start looking at some of those four million Google pages at Google I mean many of those hits come take you to the same page of course but if you sort of if you go through that you'll come to pages like this you'll see pictures of the pyramids where there's a no there's a claim made that it's aspect ratios are built somehow on the golden ratio you'll see pictures like this where golden ratios golden rectangles are superimposed on the Parthenon you see more detailed images like this there's a larger picture of some of the pyramids this claims made that the golden ratio can be found discussed in ancient Babylonian texts so these are all part of the standard story you can find these claims over and over again it's claimed that lots of modern architecture embeds the golden ratio in particular it's claimed that the General Secretariat building for the United Nations is three golden ratios stacked one on top of another it's claimed that the human body house proportions are in the golden ratio that if you measure from the top of your head to your feet and you measure from your navel to your feet it's clear that the ratio of those things is the golden ratio it's claimed that you can find it all over the place in art in Botticelli's Venus and here's a an image I grabbed from the web where there's fire to the seventh five the fifth there's all sorts of golden ratios supposed to be in the in the proportion of that it's clear that Leonardo da Vinci was enamoured by the number and embedded it in in all kinds of drawings and paintings now it's true that he was a close friend of Luca Pacioli and he was the guy that used the term the divine proportion so Leonardo da Vinci certainly knew about that number and he knew about it from someone who thought it was a big deal now the question is did he actually do anything with it in his art it's claimed that he does okay or that he did it's claimed in particular that this painter George showed us very strong claims are made about the fact that his paintings were built on golden ratios there were certainly lots of rectangles floating around it was claimed that many of them are golden rectangles it's claimed that the architect who renamed himself Luca Brasi a advocate and used the golden ratio you'll find that claim made you'll find it claimed the the Virgil composer any ad based on the golden ratio sit with the meter it's claimed that Mozart use the golden ratio in some of his music it's certainly got a mathematical sense to it it's clear that Bartok use the golden ratio and some of his music it's claimed Debussy used the golden ratio and some of his music it's clear that the nautilus shell and I actually have a real one here that that is to do with the the golden ratio it's certainly a beautiful artifact what's a so the question we're going to be asking ourselves is can you find the golden ratio in there like the answer is yes but can you meaning before they find the gold on the issue in there that's another question the reason I wanted you to bring the tape measure is we've got to have a little experiment later on to see how many times we can find the golden ratio okay it's claimed that the golden ratio crops up all over in mathematics to do with that the the Pentagon the regular pentagon the the regular pentagram that with Penrose tilings which are irregular tilings of the plane that it's can be found in regular solids that in fact all the golden ratio plays a large role there's all sorts of claims made about it in advanced mathematics not necessarily just advanced mathematics but sort of real pure mathematics well this is real mathematics and it's clear that the golden ratio can be found all over there exclaim that there are some really interesting repetitive patterns that you can get with arithmetic that if you start iterating square roots and you can play this one on a calculator with a square root button if you want you can sort of put in this thing and try the tiler iterate it through or that if you do a continued fraction constantly adding 1 plus 1 over 1 plus 1 over 1 it's claimed that that's the case again we'll have to ask ourselves if this is true or if it's just something that is clucks okay it's even claimed that the golden ratio is in the Bible again I'll just take this from the website where we've got the Ark of the Covenant and Noah's Ark okay so it's claimed that the golden ratio is really used all over the place what about the connection with the Fibonacci sequence because we've got on to this because we were talking about Fibonacci last week I mean this poor guy would be turning in his grave because he had nothing to do with it maybe wonder what the fuss was about but this was his popular legacy you know last week I tried to explain what his real legacy was so we talked about Leonardo last week and the rabbit problem and that was the problem that was in the book Libra bachi and I can't resist I'm not sure if these will show terribly well because this is a watercolor but you can more or less see it there's some beautiful drawings I'll give you a reference to this in a minute because if you've got kids or grandchildren this is a wonderful book to buy them here's the here's the garden where the rabbit goes and I guess that's a pregnant rabbit because she's knitting and then we have some little baby rabbits in months two and then we have month three we get some more rabbits and then month four we get some more rabbits maybe some more rabbits in ones five months six and so on oh I won't give you the whole 12 because I knew it wouldn't reproduce well on the screen but this is a book by Emily Gravett called the rabbit problem delightful little watercolor illustrated book about Fibonacci's rabbit sequence so definitely worth buying and we actually it's kinda book that you buy for your grandkids because you want it right there definitely well worth looking at okay and here's a graphical representation of the the problem that you've got this this growth pattern of these rabbits each one has to wait two months before it reaches fertility and then starts producing pairs of rabbits and you're counting pairs of rabbits when you do this okay so you've got one pair of arbic one pair of rabbits two pairs of rabbits three pairs of rabbits five pairs of rabbits and that gives you the sequence and if you generate it further using a computer generation you get this sequence of numbers 1 1 2 3 5 8 13 21 34 55 and so forth that's that sequence and this sequence the rule that generates it is that the new element you've gone you sort of going groups of two so you go from one to beyond the last element in the sequence so far is the sum of the two previous ones so a 1 plus 1 is 2 1 plus 2 is 3 2 plus 3 is 5 3 plus 5 is 8 5 plus 8 is 13 8 plus 13 is 21 so you just add the last two each time and that gives you the next one very simple repetitive rule this is what we're did a nice problem for Leonardo to give to get people practiced in using hindu-arabic arithmetic because they were constantly adding numbers and they're adding numbers that were getting bigger so they'll first of all have single-digit numbers and double then two digit numbers then three digit number so it's a nice way of dressing up learning how to do arithmetic so we've got a simple recurrence relation that defines this and the connection with thee with the golden ratio comes about when you start dividing successive pairs of these things so you've got the numbers one two divide two by one you get to divide the next one by three by two you get one point five then divide five by three so you divide these by pairs the large you want to the priests of the smaller preceding one and when you do that the numbers begin to converge on 1.618 by the time you get to this I mean first of all the one settles down very quickly then the first decimal place settles down to be a six then the next decimal place settles down to be a 1 when you do 21 over 13 then by the time you reach the next one this actually settles down to an 8 and so on and so on and in fact as you go out through the sequence it gets closer and closer to this at least that's the standard claim made in the books this is remember I'm still - this is still the story you'll find everywhere I'm making no claims yet as to whether these things are true or false I'm just telling you the standard story second part of this this first part of the lecture is going to be figuring out which is true or false so the claim is that this tends to the golden ratio no matter how far you go no you can't prove this by doing specific calculations you'd have to prove it by some mathematical arguments because we're talking about infinite behavior all the way out through this sequence so the question is what's true so if you haven't filled in all the questions yet I'll give you a minute to go through those 20 questions and fill in which ones are true or false and then we'll go through then we'll examine the standard story this is not high-stakes testing this is as low stakes as it gets you don't even have to show it to anybody this is in the secret of your own and by the way I've asked I've given this little quiz to very many accomplished mathematicians very few of them have got them all right in fact many accomplished mathematicians have got a surprising number one in fact he someone had given me this quiz 15 years ago I had have done very badly yes Oh just to do with the ratios between various notes and somehow when he put the music together he embedded the golden ratio in it the ratio between notes were somehow 1.6 for example if you gone they if you if you google music pie there's at least 3 or there's at least three composers who have performed of composed and performed pieces of music based on the decimal expansion of pi it sounds terrible to my ears but you can you can find it on Google in fact there was a an interesting law case a few years ago in that one of these musicians sued another one of these musicians for copyright because they said you know I was the one that put PI to music eventually that was thrown out because she can't copyright mathematics and it was said that the music was based on the mathematics now the counter-argument was that well it was all to do with the interpretation and that arguments went to very stages of the courts but the claim is just that in composing their music they're consciously oh actually it could be subconscious but the claim was that they were consciously embedding sort of building the music so that so that the golden ratio was somehow embedded in it ok ok we'll see a little bit more about it in a minute because at least one musician actually did do that at least we think one musician did okay so with there's some interesting things to look at ok we all ready to go ok so let's say let's see how you do and as we go through you can you can grade yourself right or wrong and then we'll see see what the biggest score is so the most perfect rectangle I mean there's actually two questions here first of all do human beings actually have a favourite rectangle do we resonate to a rectangle that we found the most pleasing and if so is it the golden rectangle the answer to both questions is now you know if this was a psychology class I would know and I don't know but I would know how to present rectangles to you tomorrow let's make you give any answer I want you know it depends on the on the and the way they set up on the colors are very sort of things we don't have and this is one of these standard things you can do with fishy psychology classes we don't have a favorite rectangle it depends on all kinds of circumstances so just on question one the original favorite rectangle there for the golden rectangle count with a favorite one in fact among the rectangles how close do you think you would be to the golden rectangle if you choose one right now you think it's most attractive one of those rectangles is as good as I could get it to the golden rectangle that we fooled around at the beginning to get the aspect ratio of the screen right so that this one works okay so you might like to in your mind's eye choose the rectangle that you think is the most the most pleasing if you have one so if there's one you think that one looks the nicest to you you're got one that's the golden vector how many people thought that was the most pleasing one two three four four people let's see which one do I think is the most pleasing why it's difficult I guess I would hover between that one and the one two above it which our suspect is because I have two MacBook to Macintoshes one with a wide screen and one with the smallest or maybe it's the one right at the top I don't know I've already lost which one I want yeah how do we just see let's put hands up let's see how many this one not many no on that one almost setting nobody that one now that one that one not many hands that's the winner so far above the golden rectangle no next to it yeah and I that was the one I sort of liked above that that's doing pretty well the square nobody likes the square okay right on the top actually I guess the square is top left the total squares top if next to it ah that one seems to be the winner the third along on the top well the second one another second off on the right yes yeah that was though I was hovering between those two on the top right I think is tool to world so it's it's something like that yeah my guess has always been that it's mostly habituation so you ask yourself what rectangular people look up most of the time and for a long time it was a standard television screen well things have changed a lot now you could probably distinguish between Macintosh and Windows but the aspect ratios that people like but it's certainly not the golden rectangle okay so there is some preferences but I think it's just familiarity with certain shapes okay what about this one is there anything we have to play this by the way is it's the onus is on someone who makes a claim like this the hypothesis would be that the golden ratio is embedded in this thing in a meaningful way so the person who makes that claim has to present evidence okay so the question is for which of these claims is there any evidence whatsoever that suggestive that - in the case of the fence if there's no drawings around we don't have any sketches from the people who built the path and there's no sort of strong records of these things so we have to go by just what's available so is there any evidence for this well the evidence is presented is that people take these images and they draw the golden ratio on there they go erect golden rectangles on there but you could ask yourself you know why do I pick that point I mean I can see various points you might want to put it but the most obvious points are not the ones chosen which leads one to suspect that you have you had a golden rectangle and you moved it around until it fitted some points so one problem is where do you draw these things another question is let's think about this everyone's been - the pattern is actually quite a large building now if you scale up this picture to the size of the Parthenon these rather thick looking lines are going to be huge sways in the real thing so these are not lines on the real Parthenon these are massive grid areas that you would be painting in but if you've got sort of lines that are 2 or 3 metres wide on the Parthenon then you're going to hit all sorts of points so there's going to be an awful lot of slack if you regard this as scaled up so there's really nothing more going on here than wishful thinking doesn't mean to say that there isn't embedded golden rectangles but there's no evidence for that but I'll give you a reference at the end just to the scholarly research that's just behind everything I'm saying about this this isn't just my opinion has been there's been research on this and I'll give you a rather nice readable reference that will take you to some of that research there's no evidence whatsoever that the golden rectangle was it was it was consciously or even unconsciously in the Parthenon it's just wishful thinking drawn but with these these drawings so that one's false you can tick that one I was false no evidence whatsoever so it false means unproven we're taking scientific criteria someone's made a claim can they justify the claim and if they can't justify the claim we don't accept it yeah we could stand to be corrected later on but right now this is false here's what's true euclid certainly looked at this thing he didn't think it was that big a deal because he didn't give it a fancy name he just called it extreme and mean ratio okay so he clearly wasn't thinking of this as some cultural artifact Leonardo da Vinci's friend Pacioli was the guy who gave it the name divine proportion that's got a guarantee to give this thing some you know you in 15th century Italy and if you call something divine people are going to take notice so that already gave people this sense of view this evokes numerology and religion and all sorts of stuff that's going to give it a life calling it a golden ratio or the golden rectangle that's much more recently the brother of the famous ohm of Ohm's law in 1835 wrote a book in which he called it golden so you've got words like divine and golden and perfect proportions people are going to love this I and indeed there have unfortunately that's led to an old whole cottage industry of people making claims about this thing there's no evidence that the ancient did the Egyptians knew this number at all knotting in their writings suggests they knew it and given the massive size of the pyramids you're going to be able to find golden ratios all over the place if you if you choose to so that one we're going to have to say false no evidence whatsoever Babylonians no same thing no evidence that they even knew the number false this one's an interesting one because if you talk to architects they will claim and it's in plenty of architecture books they'll claim that you're using the golden ratio in fact a few years ago I was I was the Dean of science over at Saint Mary's College in Moraga and it was in the 90s when you could raise a lot of money and I was thirty-five million dollars that's why they hired me there and to put up a new science building and so having raised the money we started to build this new science building and I was not shared the committee that hired the architectural teams now the architects came in and knowing that this was a sort of a prestigious liberal arts college most of them had a presentation that began with the golden ratio and at that point it was that was about the time when I was getting very suspicious about this golden ratio stuff in part because I just didn't buy the golden rectangle thing and I actually ran some experiments on students with the same results that we had here okay they went the way of some preferences it wasn't the golden ratio so when we had these groups of architects and they made these presentations I had a couple of questions for them and the first question was do architects use the golden ratio every one of the architects said yes question two did you use the golden ratio oh no not me but but I know a guy down the road that's another back that I didn't mean to sing of architect who claimed that they actually used it now they use things at 1.5 1.6 well that's such now it's padded by there was some light I mean three to two it is a nice ratio okay so there are some ratios that they use and it turned out that they really didn't use the golden ratio they just used simple with proportions that they call the golden ratio but it's part of this belief system that you use the golden ratio right but you can't find an architect to ever swear Jackson one with one exception you can't find an architect who ever did explicitly try to use it except instead second it turns out that in the campus at Saint Mary's College you're actually with aerial shots you can discern golden ratios that might have been deliberately put in because it was a golden ratio I mean whatever you know he was it was a liberal arts college and it's a kind of thing they might well build it we actually built something into the Science Building I built that I now regret because it's now we've now embedded in stone something I think is plenty it did I now know to be false but that's life you'll find the same claim about the Cathedral of Notre Dame and didn't mention that earlier but these claims about the architects you go on and on and on there's just no you know there's no rhyme nor reason as to why you would put these lines where you put them this your with this thing I mean you can certainly draw three golden rectangles on those three obvious vertical stacks but you have to place them very well in order to make sure it's even close to the golden ratio so that one's false okay what about this one people people people differ by the way I'm not when we use a tape measure you don't have to strip off and do this but we'll we'll do the sort of the cleaned up version of this one when you start measuring the human body there's a whole range of variation and even if you start to look for sort of the median figures or whatever you don't end up with the golden ratio this is you get close but you know one point six is a lots of things are in a ratio of one point six okay so that one we got to say is false there's just too much variation in the human body you'd also have to if you really wanted to sort of push this you'd have to give some sort of physiological explanation as to why nature was converging around the golden ratio and I'm not going to rule out the possibility that someone comes up with one of those things but it ain't happened yet and people have tried okay what about in art what you'd imagine an artist if they've read about the golden ratio might well start to embed it but surprisingly few of them actually did Botticelli certainly didn't it's just all spurious picking things wherever you wherever it's nearly the golden ratio is in the eye of the beholder here da Vinci now he didn't use golden ratios he used things like 1.5 very simple ratios so that one's false even though he knew about it because of his friend Pacioli and as there's also you get this sort of picture of the Last Supper and things you see all of the things you know the question is in all of these things there were arbitrary choices of where to draw the lines and the lines are drawn very thick so that it's impossible to give a counterclaim and say it's not even close to the golden ratio oops or drawing thick mine's on paintings is not very convincing so that we can sir you can't accept that that's not scientific evidence here's one that I found which is a delightful example well of course only some of them are golden rectangles I'm not entirely sure which one I think it's a top well in the middles the golden rectangle maybe maybe this one on its side but you have to ask yourself why the lines drawn where they are I can't find any strong reason as to why you should pick you know the guy's forehead why not the guy's nose or whatever I mean this just this seems to be just you started with lines and you do them so that you'll get some golden rectangles well that's not evidence that's just numerology that's just wishful thinking so now we can't count that one what about this one there's all of these incredible claims about this which one no evidence nothing in writings or drawings or anything nothing to suggest that there was a grid drawn in pencil with the golden ratio underneath it or whatever so we have to we have to say that's false so it's not doing very well the golden ratio so far most of these wonderful claims turn out to be completely unsubstantiated because artists did read about this and the way sometimes when artists did try explicitly to do it and I've mentioned a few people who tried to do it but that was just playing with representing the golden ratio it wasn't because of the inner aesthetics it was just cuz it was a number that had some interest there was even an exhibition called the the golden section in Paris but they just thought it was a cute name it actually added nothing to do with the golden sections there some connection because artists get everywhere but this isn't because of ascetics it's just cuz it was an interesting number they tried to play with and more modern artists have done the same thing this is one that I rather like where they just picked sort of culturally accepted representations of the golden ratio on Fibonacci numbers what about kvasir well this is one that's correct here's the architect that actually did take this seriously so we've got a true many explicitly trying to do this in designing cities and so forth and he has this this this figure this is sort of vaguely Arnold Schwarzenegger looking shaped man called the modular okay but it looks to me more like the figure in the movie Avatar than a real person and you clearly realize it in order to get golden ratios that figure had to be very stylized with a naval in a most unusual position like fungally and you have to go to those lengths if you want to have the golden ratio but it will seems the interesting thing for an artist to do look at it to do did Virgil Campos it can we found the golden ratio in Virgil now now what about music notes that Bartok no evidence whatsoever nothing in any of the writings or notes or anything to the indicated that what about Debussy the things that suggest that he might have done he might have explicitly tried to embed the golden ratio in the music um it's not convincing but it's not negative ah then we've got to leave this one as a maybe so if you said true or false for this one mark yourself correct right oh if you didn't put it in because you didn't know you can mark yourself exe everyone come back this one is correct because we really don't know what about this thing what about this this is the golden ratio in here if you go over to that science building in Moraga you will find in the middle of this hall there's a commend the men's science building that we built see the wet science building the biology building there's a big sort of world under where you come in with a spiral staircase going up what was more natural than to have a shell the nautilus shell this is this is this is marine biology in that building so there was a nautilus shell that turned into the spiral staircase I felt so pleased with that we had betted that the architects loved it it looks gorgeous it's all in beautiful stained concrete and we're going ford marble even in the not even in the 90's when there was vc money flowing everywhere marble was gonna be too expensive so it was stained concrete looks beautiful and that was a time when i was beginning to be really suspicious about this stuff you can't find the golden ratio in there even though it's on the cover of one of our books written many years ago it's true that that thing is a logarithmic spiral it's an equiangular spell this is a very mathematical object because of the growth pattern there's a lot of elegant mathematics in there it is a nice smile that grows according to a fixed angle unfortunately the fixed angle has no relationship to the golden ratio it's a different spout so spiraled as a spout as a spiral but it's not the golden spiral so it's like seeing rectangles not lots of things or rectangles but not this so the golden rectangle so it's not in the Fibonacci cell that was one of the last ones that I had to put up with disappointment okay what about these those are all true the golden ratio is all over the place you'd expect it to be here because it's involved square root of five and you've got a five sided figure so it's embedded in these and it's embedded in some of the apps some of the regulars not all of the regular solids but it's in some of them here's a rather nice one where you can get golden ratios ad infinitum remember these are these are ratios of lengths remember the golden ratio is not an actual ratio of whole numbers but if you start looking at long and short parts of diagonals and dividing them then you keep iterating and the golden ratio is everywhere in the Pentagon but of course it is it's involves route 5 and so you'd expect that to happen so that's correct is it in Penrose tiles indeed it is the the first kites and darts that logic Penrose came up with to build these things was explicitly built on the golden ratio so the golden ratio just seemed to be involved in fitting things together in an efficient way because he had to fit things together too up a plane without the pattern repeating itself and it turned out that the golden ratio was an important part of doing that fractals yeah fractals it's all this placeis thing remember factors are cob ten by taking a simple web prodution pattern reproduction pattern and iterating it for example you can start with a leaf and this is meant to be some sort of a leaf growth and you replace a leaf by two babies if you look at the tech the first one is just a single leaf then it has two children going out the top and those children are scaled down by a factor there's a scale factor going up then they have children by a factor that is scaled down and then they have children by a factor that's scale down and the scale factor is the same all the time now if you scale them down too rapidly the thing will just peter out like a tree that grows and then just sort of dies because there's not enough water so if you scale them down too rapidly it just Peters out if you don't scale them down fast enough you'll find that they start hitting each other and overlapping if these things are not shrinking fast enough then they'll start to overlap in between dying out and overlapping and making a mess you can find a pattern where things just about touch but not quite guess what the ratio is that makes that happen it's the golden ratio the golden ratio is the difference between dying out and too much growths that you overlap that's a cute one that's the beauty and here's another example of these patterns if you want to end up with this beautiful thing that looks like one of those rugs in University Avenue in those those endless book shops that we have down University Avenue the growth pattern the ratio between each of each generation in the next the one that makes is beautiful so you don't get overlapping and you don't get dying out it's a golden ratio yes is there anything there's a population dynamics population biology that was applying with that you talking about the dying out and I don't know for sure but I wouldn't surprise me and there may be people in this room that could find that out actually yeah he said if you've got a regular growth pattern the golden ratio is in there and as we'll see there are connections with biology so when you've got biological growth the golden ratio does crop up there was some really interesting ones to do with you know trees in forests that sort of spaced themselves out and things the book hasn't really been written you know that it hypothetically on that but it would be unwise to discount anything that involves biological growth it's it's in there yeah they were never touch they get closely yeah there's they say business asymptotic they get closer and closer yeah that's right in there okay and the golden ratio for reasons that are sort of understood is the ratio that makes that happen I'll hint on that at the end okay these guys look kind of surprising but if you go back to the definition of Phi in terms of that splitting up of a line that's really a different way of saying the same thing so these look kind of interesting actually this one is this one sort of explains why this occurs in biology in a certain sense this tells you that among all irrational numbers the golden ratio is the one that's most irrational now that's a very ill specified thing to say and because there are different ways you can talk about how irrational are you and this is it's not a transcendental number which is the standard way but there are ways of talking about irrationality if you're talking about numbers being represented by continued fractions we used to you numbers we usually represent as decimals but you can represent numbers as continued fractions something plus something over for something of something and most numbers the denominators keep changing so this is what this will be like a decimal expansion except it's a fractional expansion then you just keep pulling it apart and you get different numbers appearing everywhere the one in a sense when you look at the all numbers as represented by continuing fractions when everything is a one that's sort of the number that's the farthest away from everything else and so it's the most irrational in some sense and that turns out to be a pretty convincing plausible explanation as to why the golden ratio is the point where these factors are sort of asymptotic with each other so that wants to I mean those are true the the fact that these equal Phi is not deep it just comes from that equation but this is actually to do with why these things are why the golden ratio is fundamental in biology here's some interesting ones oh my goodness oh lordy yes so it's claimed that George climbing George Clooney certainly scores well on male Beauty right and I guess Paula Zahn call scores well on female beauty and so not surprisingly people have imposed golden ratios on George and Paula this one was much more recent this was actually a an advert that came up not long ago I will guarantee that's Photoshop not a powder and it's a this actually I took this from an advert on the web for a dentist he'll claim that that the dentist claimed that that he would it would alter your teeth so that they were golden rectangles and therefore make you more attractive to the opposite sex so you know if you if you're not getting enough dates you maybe should have your teeth turned into a golden rectangle it's claimed to be in drinks to do with the ear and and you know building sort of hearing aids and things so you'll find these and just two or three months ago I online I was reading The Guardian online and this was one it's in the uterus now so this was a couple of months ago in the Guardian I don't what to say I mean I I will certainly found the golden ratio in in biological systems but simply finding measurements that match and we'll try that in a minute okay again we're going to use the visible parts of the body but this is just you know scientifically so this was supposed to be two thousand years of whatever okay let's look at the four minute Fibonacci sequence we've already seen that these are related actually I better tell you yes they are related that fact I made about the ratios of the Fibonacci numbers tending to the golden ratio that's true that one actually is true it's been known a long time so that was not I can't maybe if that's one of the questions but it's it's true anyway what about this one here's a clear I didn't show you this claim earlier this may be in the quiz I don't know you have that go yeah the original quiz had about fifty questions in it I've just paired it down to one page for tonight yeah that was there's no shortage of claims there was a longer list and I can't remember which ones are left and I don't know if I left this one in a lot maybe I did because this is one of the few that's true they did do that it was explicitly built on the Fibonacci numbers only the first few because they get very big quickly but that's good enough well you just saw the Vic xqe you put it on the meter and you just you add things together it doesn't mean to say it sounds good but and it's usually the first one or two one three five and eight you just add things together yeah it doesn't make for great poetry I don't think but no I and I think to make it sound good you'd have to you know you need a sort of someone with a really good a sort of Richard Burton voice to to make it sound good someone who can take the telephone directory and make it sound good could probably make it sound good the same is true of PI music it sounds very discordant that's a yeah I mean I'm certainly well away from my comfort zone when I pass these things on but the people who study these things so yes this is this is explicitly there here's another one you know type of those 4 million hits for Fibonacci's although the 1 million for Fibonacci a lot of its to do with they're predicting stock market places and there's all sorts of claims made about this you know and the current here's one I found it was a few years ago but there's actually you get of fires all over the place here I'm not going to say it's wrong I've made it I mean I mean I've made this claim and there's been traders in the audience who said they do better than average by using this and I'll believe that because if you've got a system that you're going to use reliably and enough other people believe the same theory then you're going to get some order in the markets and you can probably make a make some money so you know with a complex feedback system if enough people believe something and a lot of people seem to believe this then you're probably going to get something but again this is this is outside my comfort zone but I I have I've seen nothing that makes me that's it's all even plausible as to why it would have any effect on the stock market other than a system that lots of people believe what about in nature there's all sorts of claims made about the Fibonacci sequence in nature Wow here's one of them did if you take various flowers and you look at their heads the seed heads you can see spirals if you take pinecones when you look at them you can see spirals clockwise and counterclockwise spirals and you can start counting and we'll do that in them and if you've enough people are bought and we'll we'll see what kind of answers we get so the claim is that if you start counting spirals in pineapples counting the spouse and the seed heads of various flowers if you start counting the the way the leaves appear on plants that you'll find Fibonacci numbers cropping up and those are all true which means I mean empirically they're true when you count them you see that and there are explanations there are scientific explanations relatively recent ones that explain why the Fibonacci numbers are care in nature so now we're into science there's evidence there's explanations for example is if you start counting just petals on flowers you won't always get Fibonacci numbers but by golly you'll get a lot of them Fibonacci numbers are very very prevalent when you start counting petals on flowers this makes this a great exercise for teaching arithmetic to young kids in school because you get them to go out into the gardens and count flowers and it turns it's a wonderful way of I mean I've been really upset if this one had turned out to be false because this is a wonderful one for getting kids interested in numbers wonderful and it's true okay so these are okay look at spirals you start counting the counterclockwise ones the clockwise ones you get a pair of successive Fibonacci numbers here's another one beautiful picture where you can see the spirals in the pine cones if you start counting them you'll find that there are successive Fibonacci numbers in these things okay and here are the numbers you can get if you start counting them in sunflowers these are the pairs wise and counterclockwise okay 21:04 three or four the one clockwise and then twenty-one compared with combined with 34 34 with 55 so you've got Fibonacci numbers pairs of successive ones ditto in pine cones due to impaired nipples and so forth okay another example would be locations of leaves on the stems of trees and plants here's a more interesting one it's interesting you've got to work on this one he started a leaf then you count the number of complete so if you go up a stalk of a plant the leaves sort of come up in a spiral okay and if you count how many complete turns before you find one leaf above the other you'll call that P and then at the same time count how many leaves are cared during that spiral so it's how many sounds you care round before you get one leaf above another and how many of these have appeared during the process and then the ratio of those two things is called the divergence of the plant I'll put a picture up in a minute spiral round this is meant to be a side view and a top view side view and a top view as you go around and so as you go around you've got a one you circle around you've got a to stick around a bit more you've got a three then a four and a five and a six then a seven and an eight and the next one will be back on top of the one okay so you've gone and you've got to eat a certain number of times round okay and if you do this these are the kind of numbers you get 1 & 2 1 & 3 2 & 5 3 & 8 5 and 13 not always but these are very very common you know better than chance okay so you know better than random for sure right okay so the question would be what's going on here this is you know you've got you've got statistically significant results here well this was known way back da Vinci who as I mentioned he didn't know about these things you just didn't put them in his paintings that was known he actually goes back to to the ancient Greeks but certainly da Vinci knew about this it's known as father taxes study of leaf arrangements Keppler noticed that the Fibonacci numbers are much more common lots of work on this done in the 19th and 20th centuries I thought one of the reasons when that when we build that science building that cent marries one of the reasons why we embedded a lot of Fibonacci stuff besides the fact that it was going to be marine biology and we wanted the nautilus shell which I still at that point thought was was connected with it was one of the professor's that sent Mary's that was there before I was there was a guy called brother Alfred Bruce oh and he founded what's known as the Fibonacci Society which is a respectable mathematical Society now has been going for many years when he retired and then died at st. Mary's it was taken over by Santa Clara University and brother Alfred Busia was also a very keen photographer of California flora fauna and so he had this enormous collection of pictures of Californian flowers and so forth which is now at the University of California Berkeley library thousands and thousands of images digital images and so it was it partly in his memory that that we embedded this in this science building but most of its okay the nautilus shell is a mistake it turns out but a lot of work was done in the 19th and 20th centuries that nailed this down because it couldn't have been an accident I mean that was just too much too many occurrences of the goal of the Fibonacci numbers so it wasn't an accident the question is what's going on to my mind this is the final part of the story the people that are in this business there's lots of eyes to be dotted and T's to be crossed the still questions to be answered but the thing that I sort of like was some work that was done by a couple of French mathematical modelers in 1993 and it really can and they sort of looked at models of growth and so forth it experiments in Petri dishes and so forth in the case of lead and divergence it was how do you optimize exposure to the Sun you know if you're a leaf you want to sort of make sure that rain falls on you and the sunshine falls on you and so the thing you want to avoid if you're a leaf is having another leaf obscuring the light so you want to move the leaves around so that you minimize the chance of one leaf obscuring another leaf and if you sort of look at the mathematics it turns out that just as with the fact Isles if you go around by the golden ratio angle then you minimize that chances that the the obscuring of one by another is minimal when you're spiraling around by the golden ratio now plants tend to have discrete numbers of leaves right I mean they have one leaf or two so when you actually start counting plants what you find is whole number approximations to the golden ratio so it's the fact that the ratios of the Fibonacci numbers converge to the golden ratio is why you get the Fibonacci numbers they're like truncated versions of the golden ratio almost it's that fact of the the ratios converging to the golden ratio so we know why the Fibonacci comes up it's because those are the whole number approximations to the growth ratio ok the God that the growth ratio which would be optimal is the golden ratio and there's a mathematical theory to explain that and the whole number of approximations are the Fibonacci numbers so that to me is a mathematician that was a good explanation I bought it hook line and sinker at that point and the same is true when you look at the growth of flowers and spirals it's optimal packing of seeds in seed heads and things if you look at the growth patterns and you can do computer simulations of this then you find that the optimal gives your eyes to these - these fouls that count with golden ratios okay and that actually that image on the top left there is a computer generated according to a growth pattern this is generated and it wasn't generated to produce spirals it's just a spirals appear because you dots are added in a growth pattern okay and going back to that point I made earlier this is really what's going on that the the golden ratio is the optimal because this number is the one that minimizes obscuring things and that maximizes efficiency in things it sort of that's the sort of mathematical explanation okay and here's why we get the Fibonacci numbers it's because they are the growth these are the approximations in terms of whole numbers to that growth pattern so we know some of it so some of it's true and we have scientific explanations and the nice thing about this is that it goes back to the ancient Greeks and to Kepler and davinci people they knew they recognized that these patterns were appearing in the physical in in the natural world and it's only more recently in the last 20 years or so that we've actually had some some pretty good understanding of why so it took a long time to come up with a scientific theory but now we have explanations okay and and so this is this is for real this is nice this is cool in fact this is so cool that we can throw away all those other things about paintings and architecture because they're nonsense oh we don't need them this is cool yeah it wouldn't be that surprising if people had embedded the golden ratio in buildings that wouldn't be that big a deal because we're people we can embed whatever numbers we want but the fact that nature turns out to be a mathematician that's cool and that is really cool and I'll leave you with one last spiral Knight equal to the golden race I'm sure you can find a golden ratio in there if you're interested in some references Mario Livio who's an astronomer there's written several books he's got a hobby in sort of writing books on mathematics he wrote a lovely beautifully research book and the nice thing about this is it's based on he went through and read all of the research literature and faithfully represents it well with one exception he actually hid it towards the end he repeats something that at the time he probably thought was true but actually isn't true so anyone who writes on the golden ratio is probably going to make one mistake and and he does too I'll leave it as an exercise exercise for the reader to find the mistake at the end but beautifully accessible book which explains well which not only explains in more detail what I've been talking about but well actually in the references lead you towards a literature okay so what I want to do now is did enough people bring these things now the hard thing about this is and depending on the pine corn you've got you might be able to see spirals this one is thus powers are actually hard to see and they're go in different directions so let's just see what kind of numbers we get some of them you can count on the bottom and let me see what I've got if you've got a felt-tip pen you might want to put a dot on some of them or even draw on them because it's easy to get lost in these things and the interesting question is if anyone can come up with a number either a counterclockwise spiral or a clockwise spiral that's not a Fibonacci number let me get the Fibonacci numbers back up let's get a nice representation of them come on let's get the numbers up so we can see it in front of us come on guys throw you can't find a Fibonacci number when you want one there we go okay so those are the numbers and I want to see if I can do man so safely the question is if you can find one that doesn't have a Fibonacci number and if you can then we'll pass it on to one to your neighbor and see if they can do a second count but it's tricky let me see if I can do a man okay you may have to hold it away from your eyes ezio got one two three four five six if you if you hold it far enough away and you maybe need to squint you'll see spirals some counterclockwise and some clockwise and the answer the way is to start counting the spirals say counterclockwise so I've got one two three four five six seven eight nine ten eleven twelve I know I've lost count felt-tip pen helps if you start up start drawing them on now I'm not going to do on this one because it'll ruin this this is my standard demo model but no this is just no this is one that I just found out where the reason that this was a kind of nice one that's why I carry this around you might find it better to do it on the size you can see the spout on the side we are from here you've got a clear spells one two three and if you mark one of them with felt-tip and then start counting them around and then go about yeah for most of these things it's easier to do it on the sides this one is a really difficult one on the sides yeah oh wow that's going to be hard to count them in there yeah yeah there's my rock to it yeah that's right yeah yeah this is a really difficult one to count such as well how would you I would count them on the side look at the side ones oh they're much cleaner is but it's crystal soit's crystals can be a Fibonacci crystal in nature so some of the Christian oh they did the yes yeah yeah yeah yeah people getting them all right we'll see we'll see yeah anyone found one that's not a criminal number it should be one of those numbers yeah and you can do it clockwise and counterclockwise no sometimes it's you but that's got really clear ones when you stand far away but you might find it easier on the sides yeah it's yeah it's really had to because the smiles are sort of epi phenomenal you have to sort of see them from a distance and it's it's not an easy thing to count yeah all right that's that that's what I would call the cancer okay yeah I will make a claim that there were Fibonacci numbers in there but I'm not going to count it find out I would yeah with kids this is grateful yeah it's great stuff with kids yeah yeah that's a great actor oops someone's book clown yeah you the spouse should be going all the way around yeah it's the ones that go all the way to well you don't have to go all the way bring you just the clear spiral so you counter spiral no no and it may not go all the way around it depends all this but you just got to count the numbers of them yeah as far as you can go yeah well that's a beauty that's the 13 your counter gates it's not easy and there vary quite a lot once you get the hang of it it gets easier yeah 13 is extremely common you either get 8 and 13 if it's a small one and sometimes you'll get a 13 under 21 5 is also gonna be a very small one so 13 is the most common yeah yeah oh let me let's see which is the best way to do this way this is like man it's hard to see it on the side so that's okay so there's one two three let me stand me put my finger on one to counter that there's one one two three five six seven eight I'm counting eight counterclockwise so if you're clockwise you probably have 13 but it's your 13 is very common I did get each yeah it's a solder you don't want to probably 13 then 13 is really really common small ones you can get five and eight that a question yeah now that we're doing sequencing the genomes of a robot are they seeing yeah don't know I'm sure if you look you'll see it the question is whether you're seeing anything meaningful I don't know I don't know um I'm always very hesitant to discount things in growth rather than it's it's an optimizer the golden ratio is is optimization and evolutions all about optimization and it just it gets in all the time josh is an insect in Oh crikey yeah yeah yeah that's good there we go he's moving around now okay I don't know insect populations no honey comes a different theory you hide those yeah I said that's a hexagon another that's a sort of a minimization of maximum volume for the minimum material yeah such a different pieces you never because there's so much urine you're mostly in for some space so you're not literally and what's actually you're not connecting anything here actually it's the force of its a ton of forces that are pushing better repellent yeah that's yeah yeah yeah yeah as I said this is a wonderful thing to give to small kids in the school right ha ha and we're here we are this is Hawaii whose website oh my hat oh yes my hat yeah she's got some nice things on this yeah yeah oh boy that this is huge in it this is probably 13 and yeah this is probably 13 and 21 so I'm sure there's thirteen one when twenty-one the other I'll make a prediction in one direction is 13 in the other direction is 21 you have to do it the other way of action so this you can you can trace someone way and you can twist them the other way and one of those will be 30 and I go and see the other one's going to be 21 because of that you can represent numbers in terms of continuing fractions and in a certain sense the one that's the most on its own is when everything is a one because when other thumpers come in it sort of brings them together it's it's this isn't precise mathematics it's sort of heuristic stuff but because everything is a one that makes it a very very special one that's keeps its distance from the other ones and because it because it's the because it's a continued faction sort of represents growth it's it's it's sort of fuzzy but it's got some some understanding to it okay so we're about to take our usual five minute break and while we're taking the break got another exercise I want to do how many people bought a tape measure oh good okay well we won't take our clothes off we won't take our shirts off however we can start measuring parts of the body and looking at ratios so there's two things you can do one is come up with interesting ratios to measure on hands and things that's about 1.6 and one of the things you realize is this is unbelievably easy to do there's an awful lot of things at 1.6 there's also a lot of things at 1.2 and 1.8 and 1.9 and 5.6 there's a lot of things to measure there's lots of sort of pointy bits and bendy bits and things what I'd like to do is if people can come up with really interesting ratios that are very close to 1.6 and then I'll knock them down and use them in some lectures ok I mean now once and this is this is empirical research for me if people can come up with interesting ratios of things in the body so you can do this either on yourself or with your partner or whoever sitting next to you just measuring things you know like this to this this to that and see interesting things that come out to 1.6 or other things that come out interesting you know I mean if you get something like 3.14159 which is pi 2.7 one you can get so we'll take a five six minute break we can use the bathroom break and put you in a break let's see what kind of measurements we can make about the body that are about 1.6 okay let's Sam let's call the the group back to order it's just this was way too successful oh okay anybody got any inch what's well first of all the quiz you all created your own quiz did anyone get them all right No okay did anyone just get one wrong anybody else just get one wrong so you win the prize this defies the supply you get a copy of my book on Fibonacci I didn't announce that first because it was always a possibility that ten people would have got a play so you got one wrong and it won't get too long two people got two and three people got two wrong I was not more prizes but I just wanted my token okay so the people up there near the top who thinks they got the most wrong no really who do thinks you got the worst one is anybody got everything wrong so who got let's say who only got five correct six correct seven hey sim you got eight correct anyone get eight off you're correct so you could go in that group eight you got eight I do have another book I can give to one of you but I want to get one book this may be very fast that to you huh you this is actually the companion book to my online course I'm giving and if you only got a to the right actually you might need that book because that's about mathematical thinking yeah okay so that's and I'm quite willing to bring another copy next week you're welcome to have a copy so I'll try to remember to bring another one next week because you may need that book as well it's called introduction to mathematical thinking and it's yeah yeah as I said most professional mathematicians get an awful lot of them wrong because it was part of our culture you know we love examples that we can use to to inter students and so we love these things you know it's like linguist talking about Eskimo words for snow I mean they can't resist it even though it's all rubbish so in it you know cos it is and are too many linguists including ones at this campus and UC Santa Cruz we can all point out the English hours just as many words for snow as such as as whichever eskimo language you want to pick anyway that's another issue measuring so anyone come up with really interesting ratios of the really close to 1.618 what's the best one we got you guys we're doing all sorts of measuring here you come up with them really cute ones sorry no no okay so that length and then that length this is about 1.6 you think yeah 1.58 yeah by standards of golden ratios that's it that's a hit yeah anything between about 1.4 and 1.8 is usually gets in with variation here reason behind it which is kind of what you're saying earlier I think about the belly button we think they be adjacent yeah somehow full so that one would fit next the other end and perhaps the digits you know parts of the thing it worked horribly yeah actually if you try to drive it by theories of growth it's really hard to get 1.6 so much more like a spurious ones where you measure yeah anyone got anything interesting to want to report on these measurements what was it that's one point one six one missed out to the handed I would guess that varies a lot on the population but I actually don't know no yeah I mean just you know one can always come up with one of these things and measure a thousand people and write a paper that says that there's this sort of number government yeah yeah yeah for people it does come out close to one point six I mean one point six is a pretty reasonable sort of racially alright that's the point of all this is if you set your mind to it and you want to show that one point seven five is important you start measuring things eventually you'll find lots of things close to one point seven five it's adding you know adding numbers after the fact I mean measuring things and then looking for patterns is you can find whatever you're looking for actually having it measuring Apple products is going to be interesting because there's a lot of user testing into what people like so data empirical data behind these things I was talking to someone about the the 16 by 9 aspect ratio of lots of things is you know when I started but the point is you can your your aesthetics change really quickly for many years I give regular lectures using for the PowerPoint or keynote slides and I got to see that aspect ratio has been very pleasant when I started doing this online course for Coursera all their things are sixteen by nine you have to reset PowerPoint to sixteen by nine all the videos a sixteen by nine I've now spent hundreds of hours working on sixteen by nine PowerPoint slides images editing videos I hate standard PowerPoint it just looks too clunky I've got used to the sixteen by nine and yet when I started it seemed too wide I've just have iterated to it so I'm now a sixteen by nine fan because I've spent the last three months living in a sixteen by nine world I've been putting mathematics ooh sixteen by nine and I've grown to like it we do it but we do a bit you eight to these things of yours so but you know that's just human beings been adaptive to whatever they're living with any more interesting numbers that are people come up with I'm just yeah it's close absolutely I mean cuz one point six is in there 1.5 1.6 1.7 those that's a nice ratio yeah yeah but the circumference of the risk to the length of doing their screaming yeah well off another one point one here yeah yeah yeah the police want you what yeah once you measure something and you get a number you can find that same number elsewhere I think you get so you get the hang of it you sort of measuring these things okay how we doing on time okay one little topic I want to finish with having having wrap that one up and this is a short one but it ties in with something we were doing last time okay and we talked last week about we got onto this golden ratio stuff because of Leonardo okay and Leonardo bring in arithmetic into the Western world if we go back this is the prequel the prequel to Leonardo took place in the arabic-speaking well those traders that were going up and down the Silk Route there were the ones that developed algebra essentially and so I thought I'd leave you with a little bit of a story about the story not a story the story about the birth of algebra okay and again we've switched them it's one of those clunky shapes again I wish this was sixteen by nine but it's not we got this um the background is meant to wither straight sort of parchment this is a nice patch mint background so let me tell you a little bit about the birth of algae because it's a very it's a story I can tell very easily in ten minutes okay first of all what is algebra it's worth noting it's not arithmetic with letters although lots of kids leave school with that impression it's actually a different I might not quite finish this but I can pick it up next time arithmetic analogy but they're really two different ways of thinking there's an arithmetic awareness an algebraic way and it was essentially in sort of 8th 9th century Baghdad that this was really brought out but it developed over many centuries almost along with arithmetic but but a little bit delayed first of all notice algebra I mentioned algebra doesn't required symbols in fact for many centuries it wasn't at least it wasn't written using symbol it was always done using symbols what everything was neither algebra no arithmetic of a guy symbols they just waited just ways of thinking so music doesn't require musical notation it's just a way of recording it and the use of symbols is not what's characteristic of algebra it's the way of thinking what you're thinking about the makes a distinction between algebra and arithmetic symbolic arithmetic the kind that we're thinking we've been talking about with the hindu-arabic numerals or other kinds of numerals that system as I mentioned really goes back to India and see year between two hundred seven hundred see II was used extensively in the Arabic speaking word in the eighty ninth centuries and then with Leonardo and various other people came across to Europe in the in the thirteenth century beginning in Italy then moving up through Germany and then westwards and symbolic algebra the stuff that we think of as algebra that's very very recent that was in France in the sixteenth century so that's that's really new the stuff that we think of as Algebra II quadratic equations written ax squared plus BX plus C equals zero that kind of stuff that's that's very new Viette ear was one of the mathematicians that that brought that out and the real distinction is the kind of thinking algebra is essentially logical thinking qualitative thinking as opposed to the more quantitative thinking that's that's really arithmetic so it's the kind of thinking that really distinguishes though this was a lesson that was certainly known in the ninth century when algebra as we now know it was really invented I'll be it not symbolic but it sort of got lost more recently because you know arithmetic what you're doing is you're calculating with numbers numbers of the things you're handling those are the things you're dealing with out because you're handling them in your mind but you're grasping with numbers whereas in algebra you're reasoning about numbers it's a high level form of thinking it's not with the numbers its reasoning logically about numbers and it's with calculating with compared with reasoning about that's the real distinction between algebra and arithmetic so in arithmetic you've got quantitative reasoning with numbers in hours you've you've got qualitative reasoning about numbers a key feature and actually this point is there's really two kinds of algebra is the kind of algebra that's studied in universities which is sometimes known as modern algebra and then there's sort of medieval algebra which is what most people think of so most people's conception of algebra is medieval both metaphorically and literally as it turns out but the feature of that was the way you thought and it was essentially introducing an unknown and reasoning logically to find its value as opposed to taking the data you've got and grinding the handle and calculating so arithmetic is essentially a forward process you start with some numbers and you combine them you add them you do various things and you get an answer algebra when it's applied to numbers and algebra can be applied to lots of other things but medieval algebra is invented essentially in any rock 8th 9th century when it was the feature was that you introduced an unknown and then you did a logical reasoning to find it's a different way of getting at the answer instead of computing to get the answer you reasoned to get the answer now that means that what most people learn to do in the algebra class which is substituting numbers into a formula that's not algebra at all by definition its arithmetic ok the algebra is deriving that formula in the first place ok so if you're simply computing you're not doing algebra whether there's a formula there or not ok for example if you're starting to solve the quadratic equation one way is how do you do it well you guess and calculate try x equals 5 so this is one way to solve a quadratic it's just it's just guess and check another way is to use the formula and plug the numbers in that's our it's make all you doing is computing the only things you're doing is computing with numbers there's another way of solving a quadratic which is factorizing it you take with the expression you manipulate it you break it into parts and then you solve it that's that's algebra your reasoning logically about it you're looking at the structure and saying we'll pass this apart into a product of two factors will manipulate it that's reasoning about it but of course is calculation involved near the end but that's that's essentially algebra it's a little bit fuzzy I'm trying to draw clean lines where no clean lines exist and algebra and analysis are certainly and medieval algebra is which still exists but I mean the stuff it was invented in medieval times is it is another way of doing arithmetic it was introduced as a way of been very efficient to do arithmetic in fact if you trace it back you can find traces of algebra back in Babylonia around 2000 BCE you look at some of these clay tablets of which there were many hundreds you find in the cuneiform writing you find what's very clearly algebra now they were actually doing geometry it was all GM essentially geometric reasons about determining linson areas of figures and in many cases as far as we can see these were just recreational problems these were done not for any application they were almost always done for specific lengths in areas but it's clear from the way it was done that these were meant to be general rules general methods for solving problems and it was definitely a form of algebraic thinking so the idea of reasoning logically in order to determine some answer goes back a long time and it was there it wasn't arithmetic it was logical reasoning wasn't computing it was general precedent patterns of reasoning about quantities and in fact they actually they didn't have a number expert they didn't have an unknown for a number but they did have in determinants for geometrics lengths so they really did have the equivalent of X's and Y's but in these cases they were unknown lines and therefore form geometric object but logical reasoning no computation it was logical reasoning with with objects so in terms of modern conceptions that's what we might want to call geometric algebra I mean I put that in quotes because that's not a recognized term I just I'm not sure whether I made that up or whether I noted that somewhere but in any case it's not a recognized term they did sometimes use this to find arithmetic solution to real problems by thinking of them as represented geometrically as did a lot of the arabic-speaking mathematicians okay here's an example of a problem from Babylonia I added the area of my two squares did this is expressed in decimal notation they use base 60 numbers I did the area my two squares 1525 the side of the second square equals 2/3 of the side of the first one and another five more okay and you supposed to compute an answer okay it was really trying to solve those two equations two squares added together the IV of my two squares are litigated 1525 saw the switch register so that's just in modern notation that's what they were trying to solve but expressed in terms of actual real squares instead of saying x squared it was a square instead of sin the second square so this kind of problem was also sold in the Rhind purpose you can find examples from Egyptians Chinese did things like this the early Greeks did things in Euclid's elements but if you look at the way they did them they it's hard to call them algebraic so in a sense the first examples of what we might with retrospect with hindsight regard as algebra was probably Babylonian but this is this is hindsight looking back and saying work and we see the beginnings of this way of thinking and here's an example and this is an example what occurs in many places above geometric reasoning where we would look at something like a plus B squared is factored in as a squared plus 2 a B plus B squared they would actually draw the figure and say a plus B whoops times a plus B a plus B squared is a squared plus a B squared plus 2 a B so they would reason with the geometry rather than the algebra but it was clearly leading towards modern algebra I mean it is algebra system geometrically the first stuff that we can see that was really beginning to look like modern algebra was in ancient Greece with Diophantus who wrote a book called the arithmetic ax actually I mean it was aiming towards arithmetic done logically you know but it was a you it was it was trying to solve algebra it's trying to solve arithmetic problems by introducing an unknown and reasoning logically so this was very recognizable in terms of modern algebra and this is around 150 250 current era he used letters official termes literals to denote the unknowns but that's really purely notational I mean it's the kind of reasoning that distinguishes algebra the qualitative logical reasoning to get answers as it happens he did use letters he also used negative numbers as it turned out which is that which was kind of unusual and he shirred have to solve equations using methods known as restoration and confrontation which sort of correspond to modern methods of solving equations you know add the same thing to both sides eliminate things from both sides whatever you do to one side do to another side things were a little bit more complicated in many cases because they didn't have subtraction so they had to sort of get around that but very vaguely restoration and confrontation corresponds to moving something from one side to the other with a change in sign and eliminating like terms on both sides and he got a bit of mileage he sold polynomial equations with powers including powers up to six so a lot of modern algebra was coming out okay that takes us up to the time of Diophantus and I'm going to stop now because we're out of time and pick up this story next week tell you the rest of the algebra story and then move on to another topic and in the meantime please keep sending in and going on to that that Google site and suggesting things because I'll try to start factoring in things that people have shown interest in yes we can talk about statistics that's already on the list someone's mentioned statistics so I'll try to throw in some statistics I'll try to meet as many requests as we can as we go through okay well I shall see you all next week and keep measuring things for more please visit us at stanford.edu

Channel: Stanford

Views: 574,181

Rating: 4.6770186 out of 5

Keywords: mathematics, logic, prove, theorem, numbers, history, equation, golden, ratio, fibonacci sequence

Id: 4oyyXC5IzEE

Length: 103min 18sec (6198 seconds)

Published: Tue Dec 11 2012