 I'm very happy to turn the mic over to Ryan Oakes who is an artist and who along with his twin brother Was actually the opening artist in our art gallery upstairs the very first art show we had here Featured the work of Ryan and Trevor Oakes. It was phenomenal. You should look them up online as well They've gone on to continue doing wonderful things and so I'm very pleased to pass the mic to Ryan Thank you Cindy for inviting me back to MoMath. It's a pleasure to be here As Cindy mentioned, I work with my twin brother. We collaborate And we're had the privilege of doing the inaugural art show where we explored our various bodies of artwork Which rub up against mathematics in a number of ways But today I'm here to introduce Doug McKenna who's going to be speaking about the golden ratio And as a by way of introduction, I'll run through a little bit of a bio of of Doug's broad career Spanning over 50 years running back to when he was a teenager So he has he's done many things ranging from digging for dinosaurs fossils Rowing a wooden boat down the Grand Canyon And starting life in New Jersey where his father was a curator at the Museum of Natural History the Frick curator for mammals Which got him involved in the you know thinking deeply about about creative explorations from an early age As a teenager he began computer programming. This is in the 1970s And and then went on to study at Yale in math and computer science And while he was there he he explored a the topic of space filling curves and the accompanying illustrations of those concepts which were executed some through hand Techniques and some through computer techniques, which was very new at the time Um after Yale, he spent some time in he spent some time in Montana But his his work with the space filling curves attracted the attention of Mandelbrot Who hired him to work at IBM? To doing illustrations of fractals, which again was very cutting edge of the time Um, so Doug McKenna drew drew many illustrations for Mandelbrot's famous publication the Fractal Geometry of Nature And many of those of those illustrations were published in the smithsonians magazine in 1983 with their first article covering the topic of fractals in 1986 Doug McKenna began Doing programming and developing developer work for Macintosh computers And worked on a developer tool called resource error. I wonder if anybody used that tool Um, it was it was critical to To macintosh programming at the time And it was so well received that That the the the tool won an eddie award in 1993 Which is the equivalent of the oscars for macintosh software Um Doug McKenna has done many other a broad range of his Cross through a broad range of disciplines Including collaborating with dna researchers at harvard more recently And he's currently working on an interactive math e-book Which is documenting and explaining his explorations and discoveries to a broader audience Doesn't stop there He's recently dipped into fashion design and has um has designed a number of silk scarves and fabrics Which are actually available in the shop upstairs. I encourage you to check those out He's working on those with his collaboration with his daughter And um today he's going to be discussing the golden ratio As he'll get into so to to lead into that I'll give you this quote By mario livio about the golden ratio The golden ratio has inspired thinkers of all disciplines like no other number in the history of mathematics Doug comes to us from boulder, colorado and please give him a warm welcome Thank you, uh happy new year or as I like to say happy or new year Uh so um And thank you ryan for that introduction And moma and the simons foundation Okay, I'm going to talk to you about the golden ratio and and Actually the art of dissecting certain simple geometries that are related to the golden ratio I'm going to say some controversial and maybe snarky things that are intended to be a funny and make you think a little bit But um some of this work I did In the 80s before there was post script and I was working on pen plotters I've done some work since and then some of it I did just in the last last year And this talk is based on a talk I did for the bridges math art conference So I like playing with mathematics and and trying to create images that are pleasing the eye for various reasons either aesthetically or Or mathematically and I kind of think about mathematical and beauty in in sort of terms of this dialectic So on the math side, there's platonic beauty if you listen to people talking about beauty There's especially mathematicians. They're they're they're all about platonic beauty and artists are all about whatever's aesthetic And it's an ill-defined word, but the thing about That the the the tension between the two in my mind is always sort of this upside down You know a catenary or parabola or ellipse whatever the curve is and there's this unstable point at the top Where there's this wonderful sweet spot balance between the two But almost everybody kind of falls to one side and that that tension is Sort of a tension between the world of constraint and the world of choice and mathematicians are always about studying constraint and the emergent patterns from constraint and And whether they start with constraint find the patterns or whether they start with the patterns and then go find constraints is an interesting Thing and then artists are about choice and making you know even even Jackson Pollock had a had a set of rules and he made certain choices, but So there's that tension and then in there's also a tension between symmetry and asymmetry And so mathematicians love symmetry. They write entire books about symmetry You know half the things in this museum are probably about symmetry But artists aren't so interested in symmetry and in fact, you know other than certain areas of architecture Most artists don't create things that are symmetric You know there are no if you look at kandinsky's paintings the first abstract painter not a single one of them is let's say bilaterally symmetric Or rotationally symmetric and why is that? Well, you know if you had a painting that was bilaterally symmetric It would be a complete aesthetic ripoff You know there'd be one bit of information for half of the painting So, you know even the Mona Lisa is asymmetric as are almost you know as are all of our faces And so so there's this tension there and the thing about the golden ratio And I'll get to explaining that in a minute is that it is This this tension between symmetry and asymmetry is sort of exemplified by the golden section so So I'll do a little bit of pretty simple math and then the rest of the talks can be about pictures that sort of embody Pictorily some of these mathematical concepts because that's just you know, that's where it's more fun to explore in my mind So what is a section a section is when you take a Something that you measure a foot You know a ruler a yard stick or you know a mile and you want to cut it into two pieces at some point So you choose a point you call it x and you say, okay, we're going to divide this at x So you can choose any x you want you can choose a quarter and get And you know and the other pieces three quarters or you choose a third and two thirds or half and a half That's right in the middle. That's a symmetric Choice you choose a ninth and one tenth and so forth and in mathematics mathematicians like to go the go the distance and they'll even talk about Dividing something up into two pieces one of which is non-existent So you can have x equal to one or zero and so that turns out to be useless in the real world But it's elegant in the mathematical world. So we're choosing this Choosing this x and it can be any it's a measurement It can be a real number anywhere between zero and one And so the question is as well, what what can you do with that? Well, you already know how to add them together because they add up to one. So what about dividing them? So let's create the ratio of these. What does that mean? Well The distance of the smaller pieces x minus zero so that's just x and the other one's one minus x So what's the ratio? It's x over one minus x so you can see a pictorial fraction there if you will on the right Um, so let's plug in some numbers see what happens if the x is zero the answer is zero if x is A fifth point two the answer is point two five which is a quarter if the x is four tenths the answer is two-thirds If x is five well I'm sorry point five Then in point five over one minus point five is point five over point five the answer is one So you can see that x is growing and if you put in higher and higher numbers It blows up until finally you get to one and you've got zero in denominator It's undefined so we say well, that's infinity so you can graph that and you get Um, you know the simple graph of this formula x over one minus x and now if you graph The value x equals r you notice that the only intersection there is at zero So if x is zero then the ratio is zero and so there's not a lot of There's no real constraint between the two because there's no no other value of x that's kind of distinguished there When you equate the two But remember it's not just two it's not just two Two lines you have you actually have three quantities here You have one and x and one minus x So let's think about a really simple thing you can do with these three quantities with respect to ratios and the answer is Well, let Put them together this way Or the hole is to the greater part as that part is to the smaller part And the thing is is you ask yourself well, well, okay I don't know what those numbers are but maybe they could be equal So you think about that and then you make yourself a little equation So you have one over x is the left hand fraction and it equals x over one over minus x on the right You want those to be equal on that if you look at that you immediately notice playing around with it that if x were point five It won't work because if the left hand will be two and the right hand will be one and two never equals one What that tells you though is that the solution is not symmetric and that's interesting It's a very simple declaration of a constraint doesn't have a symmetric solution So let's do a little bit of math here You see you have the one over x equals x over one minus x and you can simplify that to the next One minus x equals a squared put that all on one side You've got a quadratic equation and x you just use the quadratic formula to solve it So you see that x is equal to minus one plus or minus the square root of five all over two So there's two solutions mathematically one is negative And we're not going to use that because we're talking about measurement here and a negative measurement doesn't make sense Especially since we've constrained ourselves between Numbers between zero and one but the other solution is point six one eight o three three the golden section And it's an irrational number because the square root of five is involved and so those digits go on forever So everything we're talking about here is essentially an ideal. It's a it's an approximation to this this number that's a little bit higher than Three fifths right point six Well, that's called the golden section and I like to think of these as disections because it's two pieces And the whole is to the greater part as that part is a smaller part. Well What's the reciprocal one over x well if you put that in a calculator you say one over point six one eight Oh three three dot dot dot and it turns out that the reciprocal is one point six one eight oh three three So the the infinite number of digits are exactly the same it's repeated So that means that the reciprocal of this number point six one eight oh three three Etc is one more than the number itself And that's a unique property and it's uh, it's true of the reciprocal and vice versa So that's kind of an interesting Mathematical property of the golden golden section And we call this reciprocal the golden ratio and it's kind of think of it as an improper fraction Where the larger part is above the in the in the numerator and the one is the denominator So it's equal to one point six one eight oh three three Now in math the golden ratio is represented by the greek letter phi Sometimes by the greek letter tau, but we're going to use phi because it's pronounced phi or also phi And i'm going to play some word games with that in a minute This number has captured human imaginations like like ryan's quote from area livio since the ancient greeks And the reason Well, well before we get to the reason Where in this room? Can you find examples of golden ratio use and i'll give you a hint it has nothing to do with the museum of math It doesn't have anything to do with the uh the sinks in the bathrooms, which are pentagons It doesn't have anything to do with the various tilings and exhibits that might be pentagon based or so forth So where do you think in this room right now? You can find examples of the golden ratio Any guesses anybody? Yes Hmm Well, there's a theory about that, but it's kind of bogus Yeah Pardon me I don't know. I haven't measured it. I doubt it, but uh, that's possible, but it but that doesn't count because that's part of momath So where in your own pockets can you find the golden ratio? all right If you have an iphone or an ipad Apple tells people who develop software like me for the ipad how to design icons and they they give you this grid To lay out your icon imagery in a very simple way and the grid starts with a square and you inscribe a circle in it The circle is the largest square you then apply the golden ratio to get a smaller circle You then do it again to get a smaller circle Now you can't just go backwards from that because if you undo it just get back to the same circle So now you cheat a little bit and you increase it by the square root of two Now you can get back to the golden ratio and you get to that circle, which is The inner circle times the golden ratio You then draw these lines out to the corners and these corners right here Where are where the arcs of the rounded corners are supposed to go through? So that's how app all those icons on your iphone are based on this application of the golden ratio And if you look for instance at the settings icon on the napple design Those are three concentric circles which have radii based on the golden ratio Now why is why did they do this? Well because designers and you know Various other people have for a long time worshipped the golden ratio as some sort of ideal and they they think that it They must be doing something right if they're using it and I call this faith-based design And The thing is is if you use the if you use the the numbers 1.6 and 1 by 4 They would work with Almost precisely the same aesthetic results there you wouldn't be able to tell the difference So what they're doing is there it's an homage to an idea or an ideal But the real goal of a design layout grid is not to use the golden ratio It's to create a standard so that all sorts of icons have a sort of a comparative look and don't you know Use the similar amount of space So the golden ratio, which is also has been called the divine proportion is not And you know, we need to be very careful about you know issues of sacred geometry and so forth where people As people always do like to impart a little a little too much meaning to some of the things that maybe they don't understand as well In any case, let's look at the golden ratio squared Okay, so the golden ratio is 1.6 180 through 3 dot dot dot And we know that the reciprocal is one away so you can subtract one to get the reciprocal And now you multiply by the golden ratio on both sides So you get this equation, which is the difference between the the second power and the first power is one so we get this Identity and so we see that the square of the golden ratio is 2.6 180 3 3 3 So it's the same repeating digits So it's exactly one more than itself when you square it now That's interesting because this golden ratio identity. It's the fundamental golden ratio identity is I think of it as the sweet spot between additive and exponential growth on the left side You have exponential growth on the right side It's additive and so there's something balanced about this even though it's sort of this asymmetric quantity So if you look at the golden identity and then assign exponents to all three parts You see that phi squared equals five to the one plus five to the zero five zero is one That tells you that any power of the golden ratio can be reduced to a simpler expression in just The golden ratio itself without any exponents except one so forensic So if we take the the golden identity here and we Multiply it by phi we get phi cubed equals phi squared plus one and that turns out to equal two phi plus one You can take phi to the fourth and it's equal to phi cubed plus phi squared Which is turns out if you reduce it, it's three phi plus two you can go the other direction Here's the you know what we saw before the reciprocal is Is one away from the first power And you can go up and even say that one is equal to the reciprocal of golden ratio plus the reciprocal of golden ratio squared So what's interesting about this of course is these integer coefficients, which are the fubinacci numbers And the fubinacci numbers is huge, you know, there's a the fubinacci orders are a growth pattern that are It's like the discrete form of the golden ratio And there's all sorts of relationships between them that crop up in zillion ways And there's you can read about all all this on the internet What's interesting here is that these are the fubinacci numbers also except that there's that funny one up in the right And that's actually the minus one fubinacci number because fubinacci numbers go backwards as well as forwards And so you get to this Experimentally you get to this identity I'm sorry where the nth power of the golden ratio is equal to the n minus first plus the n minus second Which reminds you of the definition of fubinacci numbers Which is the nth fubinacci number is the sum of the n minus first fubinacci number and the n minus second fubinacci number But the nth power of the golden ratio equals the nth fubinacci number times phi Plus the previous fubinacci number Well, the greeks knew all about the golden ratio geometrically And one of the things they figured out was that the ratio of a diagonal of the pentagon the regular pentagon To the side if the side is one is the golden ratio and they thought this is uh, you know This was sort of interesting. They didn't they didn't create numeric versions of it because you know back back then you know people were dying because they were arguing that the square root of two was irrational and uh But look at this, okay The golden ratio occurs in an equilateral triangle inscribed in a circle Turns out that the if you just extend this side of the of the inner triangle to the to the arc of the circle That triangle cuts that that line into a golden ratio. That was discovered within the last 50 years. It's amazing It's so simple And it was discovered by a mathematical artist, which i'm happy to say So where else does the golden ratio occur? Well, this is a famous definition of the golden ratio It's called a continued fraction where the Denominator of this fraction is a sort of a copy of itself and it goes on forever And this is the simplest continued fraction and why is it related to the golden ratio for the exact same reason that we just talked about The the denominator of that fraction is the same expression because it goes to infinity So they're essentially equal. So it's basically the same as the golden identity Phi equals one plus one over phi or phi squared equals phi plus one all right, so Why does the golden ratio occur in these different ways and the answer is not because it's divine not because You know it's magical is not because we should this and the other it's because it's simple it's really simple And so there's only so many very simple constraints in mathematics and this is akin to a An adage in mathematics called guys strong law of small numbers And guys strong law of small numbers started as a joke But it's really a truth which says there aren't enough small numbers in the world For all of the uses we need to put them so they crop up in different ways So that's you know Think of all the places that you see the number two. Well, it's it there's a lot of really Good uses for the number two, but there's only one number two And the same thing is true with constraints in mathematics There can be you know There's only simple some simple constraints like adding two numbers together and getting a third number and then Putting that into a feedback loop. That's very simple So that's the real reason that the golden ratio crops up in a lot of a lot of different ways So in the pentagon, let's look at the pentagram, which is all of the diagonals And the golden ratio happens all throughout here So we saw that the diagonal to the side is equal to the golden ratio, but you can find it in parts of this So the the side to the portion portion of that diagonal and you can use similar triangles to prove this is also the golden ratio And the side of that to its base is also the golden ratio So it crops up in all sorts of ways and you have these two Quantities you're making a ratio with and if you have two quantities of two lengths, what kind of triangle can you make from it? Well, if you only have two lengths, you can only make isosceles triangles So it turns out that there are two triangles that are very very golden and One of them has a short side and two long sides. One of them has one long side and two short sides And they're made up of sides and diagonals of the pentagon So these two golden triangles, uh, you know, if you want one's bigger than the other So you can actually do the math and figure out that it's exactly phi squared an area larger than the smaller one And these two if you take pairs of them and glue them together You'll get what are called penrose tiles and these were first described in 1970s These can create Non-periodic tilings that are quite beautiful and have all sorts of engineering Structures to them and they're a little bit related to some other things I'm talking about but you know, there's this huge literature about them and lots of good pictures You can find and I say rediscovered because Pentagonal symmetry may have also been used in certain islamic designs, you know 500 years ago and that's been a subject of research in recent years So the penrose tiling creates aperiodic tilings. They don't repeat themselves the way that square tiling or triangular tiling or hexagonal tiling might They can in some sense, but but what's really cool about them is that they create these Aperiodic tilings where where you can't translate the tiling in any direction and have it fall right on top of itself well the thing about the two These two triangles that I like and it was interesting was the fact that they have a codependent relationship Or a core cursive relationship the diagonal of the pentagon that crosses these cuts these two triangles into two pieces And look at those two pieces those two pieces are identical in aspect ratio to the same two pieces We started with so a can be carved up into two triangles one is a smaller copy of a one is a smaller copy of b Or actually it looks like it's an exact copy of b And then b can be carved up into two triangles one is a smaller copy of a and one is a smaller copy of b Now there's no other triangle that you can do that with but It's really kind of beautiful Now the interesting thing about that dissection though is that you have one bit of freedom You can choose to carve it up from the lower left to the upper right Or from the lower upper right to the lower left and do that in each triangle So if you're going to carve triangles up into smaller and smaller sub triangles you have this freedom In any whatever algorithm you use to play with those those symmetries So let's do an experiment. Let's start showing what what that looks like. So Here is the taller of the those golden golden rectangles and we're going to divide it up into two And we're going to divide each one of those up into but i'm making a choice here because I have symmetry choices I'll choose that now. I've got four of these triangles Two of each and so forth and we can continue this on every time we double the number of triangles And obviously you wouldn't want to do this by hand. That's what computers are great for particularly since they can model geometric Numbers and constraints pretty accurately. So let's keep going until we have I don't know how about 120 000 golden triangles and And you start seeing these incredible Textures and what's going on here is interesting And it's and it's cropping up in a very sort of mysterious way because the golden ratio has this balance It's the balance between You know additive and exponential growth It's the it's the balance that keeps all these triangles and we can go and look in here And we'll see that all these triangles have the same shape It's the exact same set of triangles if you use any other ratio You can carve up a triangle in two parts lots of different ways But if you use any other ratio, you're going to get a an unstable Drifting away from the from the same triangles. And so there's a there's a balance going on here, which is really interesting Let's try doing that in a different way because it turns out you get different textures so we'll Carve it up this way And again, we'll go up to maybe 120 000 triangles and so we get a different Texture golden texture. So what's happening is the darker parts of where the smallest triangles occur because you're using a fixed line width to to To illustrate this so the smallest wrecked the smallest triangles are going to have the most black next to them So that's why there's this variant. It's you know, but you're seeing something It's like looking up at the Milky Way in the in the in the in the sky or you know, you know Or maybe we're you know, maybe it is religious. Maybe it's like the shroud of Turin and you know, whatever And so but that's just you know, yet it let's try another another another tactic. Okay, let's do this Try this dissection and we'll get a different texture and this is I like this is this is kind of intriguing You know, you get these weird patterns and and wonderful, you know it's all self-reflectional and this it's fractal and self-similar and so forth, but Yeah, it just has a texture. I think it's lovely and and and it's balanced It's as balanced as it can be if it's going to be asymmetric and non homogeneous So let's try one more And this one is really kind of interesting also What happens here and what happens is all the small triangles seem to be hewing Wait till it draws hewing to the the larger subdivision to the line So it's sort of all this convergence is going to the original subdivision lines And you know, you can zoom in and and look at this Oops Now this is why it's nice to have computers to draw this because you know, you would not want to do this by hand Um, now there's another fun thing you can do Which is when you subdivide Let's carve this triangle up into three triangles and then ignore The the shorter one at the top and then subdivide again the two and we're going to leave all of those The the B triangles unsubdivided. Okay, and so when we do this You actually end up getting this ascent this fractal, which is um, you know, you can zoom in and it's kind of a tree-like structure And there's It's composed completely of just the a triangles All of the B triangles you're not subdividing. So they're elsewhere And it it you know, eventually it has zero thickness And now if you use a different symmetry And do it again Now all the A triangles converge to essentially a snowflake curve, which is the pentagonal snowflake curve and and you can have fun with color now and and For instance start coloring in the the triangles. You're not subdividing and use three colors and then rotate them And so you get these converging stars Let's show you, uh You know, uh This it's a sort of a growth pattern into this fractal And if we do the other symmetry we get this So these stars are converging to different fractals, which is kind of interesting. They all have the same fractal dimension And uh, I just think that's you know, it's just lovely and and of course it's there's no, you know, I mean, there's local symmetries Oops, but it's not it's asymmetric and that's what makes it interesting and aesthetically interesting as well as mathematically interesting to my mind All right, so You can generalize the golden ratio There's things called silver ratios the golden ratio is a special example of a silver ratio silver ratios exist For all odd-sided polygons just like the golden ratio exists for the pentagon the silver ratios for the heptagon the non-agon etc So for the heptagon the seven-sided Regular polygon there are three Co recursive triangles and each one can be carved up into a similar copy of one of the three When you put them together in three different ways And I I played with this back in the 80s And this is before postscript existed in graphical languages and I was doing pen plotter drawings of this stuff with repitograph pens and and and paper and trying to create these huge dissections and and um So it was a lot of you know, it was really interesting and I found new fractals and so I I did this I found this particular fractal which is embedded in the uh The tall triangle and you can find it in the others as well It's I called it a heptagon because it very it looks very much like a dragon But it's it's like the dragon curve in a sense. It's got a fractal boundary and It's got self similar parts But you can look at look at it this way and you can see that I've chosen to illustrate it in a very dramatic way And I I think it's a successful enjoyable thing to contemplate and look at and it's got a lot of detail And I also did another experiment with these This triplet of the the heptagon triangles and I called that a flur to lips And you take a circle and you divide it into 14 parts and then those create triangles And then you recursively subdivide them using the three triangles and play with Certain triangles that you don't want to subdivide and then you play with color And then it turns out that if you do want to do it and then you've got the asymmetry and I just think it's really pretty and fun fun And you can look at it. It's just a myriad set of triangles three triangles all the same shape just different sizes Well, okay, so let's go on now this golden ratio had these two pieces and we used them to make a triangle out And it turns out it's pretty elegant But what else can you can you make a rectangle out of these two pieces? And the answer of course is yes And it's known as the golden retro or as I call it the so-called golden rectangle Now I'm going to say some things this this has been worshiped for thousands of years The you know, there's this theory the parthenon is based on it But it's not really true and you can find it embedded in the icosahedron or the dodecahedron and the golden ratios and You know the four-dimensional dodecahedron and and and you know There's lots of places and there's this theory about art frames being the best when they're the golden ratio but I'm going to prove to you that It's really at best the goldenish rectangle And in fact, it's kind of miserable So I really I tend to call it the miserable golden rectangle and I it's not I'm telling you it's not I'll explain. Okay. So what is it? Everybody who's in Oz about with the golden rectangle? Well, you can carve off a smaller golden rectangle on the right side and If it's golden the leftover or the nomen to use the Greek word the leftover when you take a self-summon part away from something Is the unit square? Well, okay, I'll grant you that that's the only rectangle that can be true for and that's based on the the fundamental identity but You can do that again So you carve another one off and it's a ratio You know another another golden section and you're left with another square and you can do that again So you get these squares that reduce in size by the golden ratio each time And I think you can do these golden ectomies forever and obviously there's something interesting going on If you do this add infinite and there's some point of convergence there And it's at the the intersection of these two diagonals So you can actually very quickly figure out what the coordinates of those diagonals by just summing two infinite series of squares The white square is summing to the right and the blue square is summing upwards And you find the coordinates are you know 1.17 and 0.72 which happen Well, they don't happen to be of course their ratio is the golden ratio well There's another cool thing you can do that shows you that you can divide the golden rectangle up into four Pieces that are all the exact same shape only in the limit They're an exact same shape in the limit and that's I'll admit that's kind of interesting But they're not so some they're not similar to the original They're sort of it's an artificial construct Well, then there's the the golden spiral everybody hears about the golden spiral and and you know, it's basically based on Putting a spiral into this set of reduced squares and if the problem is of course is that's not the golden spiral I just fooled you Okay, my presentation software only draws circular arcs. I doesn't draw logarithmic spirals It looks like a logarithmic spiral. It's real close to the logarithmic spiral But it's not and this has fooled lots of people for a long time For instance, there's all sorts of talk about the golden ratio You know being implicit in how a nautilus shell grows. Well, yes, a nautilus shell grows exponentially But it's not really clear that golden ratio has much of anything to do with at all And there's a paper about this that was done by yet another mathematical artist a couple years ago published in in the bridges conference and you can find this on on On the web at their archives and it's a it's a great explanation of the bogosity of this Of this theory that the golden ratio occurs all throughout nature So here's my complaint from the you know from a self-similar recursive subdivision point of view Like we were doing with the triangle the goldenish rectangle. It's just second rate You know, it's just unworthy of all this attention that's so steamingly heaped upon it And i'm going to prove to you that it's completely useless. Okay, so the problem with the unit square is there's nothing golden about it That's worth recommending at all There's nothing golden about a unit square. It's only you know this this piece you took off So the golden rectangle is at least 61.8033 useless, but of course You can do the same thing again and now we add the uselessness and we get up to 85 percent and we add the useless now We're up to 94 percent Now you can do that forever and in fact mathematically you can compute the limit of all those sums of squares and It's the golden rectangle in completion except for one point. That's that one point where it converges That's the interesting thing. Everything else is useless all right So something Something is deeply wrong here. What you know, what happened all the elegance everybody's been saying the golden Rectangle is just awesome. And it's we've been around for thousands of years, etc It's like being sold a fine steak that is over 61 fat and having to feed it to phyto The problem is that we've been shoehorning the golden ratios one dimensional elegance into the wrong two-dimensional rectangle So let's get back to first principles What's the question? The question is how many ways can a rectangle be subdivided into n self-similar sub rectangles well If you have n equals four, it's pretty obvious that works because you're basically carving a rectangle up into two halves And they're all four of them are similar to the whole they're each half size So that's pretty obvious that's trivial And if you think about that then it works for any n that's a square of some integers So it works for n equals 16 or n equals 25 or n equals 100 and they're all similar to holes So you can solve it for n equals a square number. But what about a simple number like two? What about three? For two there's only one distinct way in it's the one by the square root of two rectangle And you can cut it in half and the two the two halves have the same aspect ratio They're rotated which is interesting and in fact, this is the basis for European paper sizes So the a1 a2 a3 and a4 paper sizes are all based on carving rectangles up into two pieces by the square root of two This proves that the europeans are much smarter smarter than we are because when we carve pieces of paper up We have these extra leftovers and the aspect ratio changes, but they've got a really elegant thing going over there Well, what about three? It turns out if you go through sort of all of the possible ways you can carve up rectangles There's three distinct solutions And what do you think the aspect ratios are? well any guesses so If you think about what I just explained with the root two obviously there's one by root two So that left one is one point four one four two That's the square root two the middle one if you know your math to recognize that that's the square root of three And then there's this number one point two seven two. Oh, what the heck is that? The answer is the square root of two the square root of three and the square root of the golden ratio So here's this rectangle. That's one by the square root of five one point point two, which is Turns out extremely golden And I consider this the better golden rectangle and so i'm going to call it gr for golden rectangle It's one by the square root of five So on the lower left you have it and we're going to what we're going to do is we're going to scale it up We're going to multiply it on both horizontal and vertical dimensions by R r square that's the golden ratio square which is equal to r to the fourth So it turns out to be an r to the fourth by r to the fifth rectangle Now what does that teach you? Immediately you can see that r to the fifth since it's a rectangle has to be equal to r to the third plus r Plus r and immediately you see that r to the fourth Has to be the same length as r squared plus one Well, one of those guess what we're right back where we started to the the fundamental golden ratio identity because you can divide those Equations those identities in r by by the square root by r and then just take the square r r squared equals equals five And then here's the fun thing it works for areas as well as linear dimensions So this is our fifth. What's the area of the whole thing? It's r to the fifth times r to the fourth What's the area of this? It's r to the fourth times r to the third So the area is r ninth of that of the whole this is r to the seventh This is r to the third and this is r So you have that identity So you can say r and and you can simplify that to five to the fourth equals five third plus five plus one And that simplifies down to the usual golden identity So that's kind of cool too because that's you're gonna do you're You're getting the same identity in both one dimensional and two dimensional areas But now every part is useful for recursive subdivision just like in the triangular case But you have four choices of symmetry here You have essentially each of the four possible ways you can divide this golden rectangle up into three parts that are self-similar Are equivalent to different origins different coordinate systems at the four corners And that's really useful when you're writing say postcript programs because it's very easy to do these transformations in postcript so That means because each of these are three different Independent rectangles that means that there's 64 possible ways that you can Choose these subdivisions. So you've got a space that you can explore And you can't visualize it very easy. You don't know what's going to happen. So let's look at another demo Let's do this down. Maybe 16 000 times So here's 16 000 golden rectangles and they tile the original golden rectangle and they start creating Textures and some of them are more interesting than others. So let's Scroll through some of these 64 And you see that you're you're getting groupings of the smallest into interesting structures And suddenly you start seeing emergent patterns and You know where there's maybe some spiraling going on and it's all based on this tree of linear transformations that are positioning on this myriad set of Shapes in different sizes all the same shape in different ways so It's asymmetric and then, you know, suddenly this is looking like it used to be straight lines but now it's gotten kind of wavy and You know, you this looks like it's sort of a gasket a fractal gasket and you can go on and and see these textures Then suddenly whoa This is It's the same rectangle, but now it's rotated a bit. So how can all these little tiny rectangles that are all You know, right angles create these larger rectangles that are at a diagonal. That's that's kind of interesting And then this one is really weird because you get the same rectangle You can sort of see this rectangle there But then it sort of deteriorates and just melts away in these areas. It's like it's it's like it's rotting It's very strange and so You know, what's going on is this these patterns are interacting with your visual system and they're firing neurons in your brain to To remind you of you know, this one reminds me of for some reason I keep thinking of simies cats when I look at this I don't know why and then you get sort of these fractal structures and You know, they're all recursive and self-similar and then you but you get these emergent lines and They're related to the golden ratio and the angles and and it's wonderful that these crop up because you're you're discovering the patterns Before you're discovering the constraints. I really like that. It's you know, this is a that's a really strange and interesting pattern there really unexpected And you know, they're just lovely and you know, like this one is obviously very organized from the largest to the smallest Golden rectangles, so it's sort of this sweep of getting darker towards but it's all hierarchical And in this one, I really like a lot for various reasons and you know, it goes on and on So I did this piece called fiber space and it was based on one of these dissections But then at the last minute I took away all of the golden rectangles and I substituted a different rectangle that was sort of inset and it wasn't a true golden rectangle, but I widened the lines. I used colors and I superimpose them on each other and amazingly enough by choosing just slight variations I came up with a design where You see nothing but hierarchical set of spirals. It's like if you actually look at this there are no spirals in this picture whatsoever, it's It's it's a you know You've got this tree of linear transfer transformations that creates the set of spirals implicitly, but then when you actually Impart some aesthetic choices or just play and they pop out at you I really like this. My sister's going to make a quilt out of this, but in case You know when you look at this you'll see that there's a whole Whole slew of different sizes of golden triangles from the largest in the lower left Down to you know the little minus down here and then there's the largest So it's an interesting question is why have this tree of subdivision be all the same depth? So that introduces the idea of lag subdivision So and what that does is it creates a a more homogeneous set of these tiles In the subdivision and if you look at the Area sum you say well let's subdivide the larger ones More than the smaller ones that should work And because of the identity what we what you can do is you can subdivide the largest one Twice to twice the level as the medium sized one Which is one times the level of the smallest So and then when you're doing that let's scale the picture up by R squared each time which what that means is that the the smallest triangles in these resultant Subdivisions are going to be one by r at every stage Let's look at this. So here's the first level subdivision And do it again, but we're not subdivide We're not subdividing this one and this one yet, but we did subdivide the big one here And notice we've got two two rectangles here. They're the same size And we do it again And we've got two over here the same size now finally we start subdividing on this side because You're you're creating a only a set of four of these golden golden triangles the tile the original And you keep subdividing and because of the symmetry that I chose here You get these patterns, which are really kind of remarkable because they're not symmetric They're a periodic They're completely regular in some sense And you can't see where the original subdivision is anymore The original subdivision is there, but you're getting this emergent structure out of out of this simply by introducing this lag time So let's try it again with this. This is a different symmetry Now what's interesting here is now you've got a bilaterally symmetric subdivision and it stays bilaterally symmetric up until It an asymmetry is introduced So it's asymmetric for a little while and it goes back to being bilaterally symmetric And as you go down deeper and deeper you get what I call the golden plaad design and um, and if you look at these Bars, there's two here. There's two here, but then there's one then there's two and so forth and there's almost certainly a relationship to uh, what's called the the What is it called the the The golden I can't remember drawing a blank, but there's a sequence of ones and zeros It's called the Fibonacci number and uh, it's it's probably related to that because of the way that these linear transformations work Let's look at the counts of these rectangles. Okay, so there's one tan. There's no purples. There's one blue There's one sort of maroon And that those those numbers correspond to coefficients in this uh expression in the four powers of the golden ratio So five to the fourth remember we scaled this up is equal to five third plus five plus one And five to the fifth is equal to five to the third plus five square plus two five plus one Etc etc etc So you can get all the way up to say five to the fifteenth equals a hundred and sixty nine five q plus 104 etc Well use the golden identity to simplify this Okay, and you simplify this down to five the fifteenth it equals this expression just in the golden ratio And then you combine like terms and you get five the fifteenth equals six hundred and ten five plus three seventy seven And if you know your Fibonacci numbers those are Fibonacci numbers the fifteenth Fibonacci number and the 14th Fibonacci number So that's a visual proof of this identity that the nth power of the golden ratio is a The fifth the nth Fibonacci number times five plus the n minus first Fibonacci number So there's five to the 18th right there And it equals You know those two Fibonacci numbers etc So these are the We'll scroll through some of these each of these is a different one of the 64 symmetries and each one creates an a periodic tiling And you know some of them are just sort of right. It's strange They're homogeneous. They're sort of random and then you you come up with this incredibly organized tiling except you notice There's a bar down this side and a bar along the top That's half the size of these And it's the same set of tiles They're just being organized by a different set of symmetries out of the 64 possibilities So you get these diagonal lines that come in pairs and singletons and And you know, it's much more homogeneous It's not it's not as textual as the previous ones because there's only four sizes of tile here And yet they tile a copy of themselves in the large scale So this is just an exploration of structure and there's more plaids here This one it looks like it's bilaterally symmetric And this this one's kind of it's kind of clumpy and sort of You know, there's sort of local symmetries here and there and you get That another plaid and sort of a choppy bunch of lines this way And so it's just you know It's this fabulous and and here you get some large scale structure where you you see that there's sort of a line like this Wait, where is it? No, it's a I can't see it from this angle very much But it goes up this way you can sort of see that there's some large scale structure going on there And that one's kind of that's really kind of interesting where there's this clumping going on with like triangle like like rectangles, etc anyway, um Let me finish up and show you what last thing this is related to which is What's called the almond almonds be tile and it's derived from the the better golden rectangle and this tile has a property that With it and a copy of itself mirrored and smaller it can tile the plane Just like squares can't accept. It's completely a periodic Where did it come from? You know when almond described this in 1978 Um, he didn't explain it very well. These are his notes And he was interested in what are called reptiles, which is figures that can be carved up into copies of themselves Like the square So he sent these notes to martin garner who ran the mathematical games column at scientific american and in february 78 And he says oh look the square root of the golden ratio and he just drew this ab initio and it doesn't explain where he got it from well it's the goal if you take away c And then subdivide a once And rearrange and glue the pieces together differently you get the etymology of of almond's tile This is a reduced copy of this by r And together and it's also a notice. It's a mirror image And so together the two create a a recursive subdivision, which is The moment I thought about that I said oh well, I know about those In any case so gardener And it was almond wrote this to gardener He said you know these are reptiles and their boundaries can have they can be related to fractal curves remember This is only a year after mandelbrough Had his first english book on on fractals out in 77 And the part about this i like is where he says i believe these tiles would be a good subject for Art and would be produced and would be produced would have produced them by computer myself if I had access to an x y plotter This is 1978 now almond was very eccentric. He probably was I don't know. I think he was autism spectrum he Died in obscurity. He didn't give lectures. He just wrote this stuff And so martin gardener wrote to solemn golem who was invented the term reptiles and also Pentaminos if you're familiar with those he says this might be of interest to you I've also given a copy to mandelbrough He's some sort of genius who moves about from adres to adress in massachusetts and seemed to have no fixed means of livelihood He hasn't told me anything about himself beyond he has done Sort of sort of trading in computer science from time to time He sends me results on this and that the enclosed being the latest well Two years later. I ended up being mandelbrough's one of his several Programmer illustrators and been one mandelbrough gave me these notes from almond. I said, what do you make of this? And I looked at it and my head exploded. How could he figure this out? Where does the square root of the golden ratio come from and and then we moved on to other things We actually never illustrated from it or or explored those fractal curves that he was sure were there And then a couple of years later I derived this for myself just trying to say well what rectangles are self-similar and then I realized this was related to the almonds be tile and I had these notes that I still had and so I did this Originally as a plotter drawing and it's got about 40,000 golden bees in it You know, it's maybe this wide and I've done it. I've done it five feet wide on a Later printer not a pen plotter And this I call golden omen because no man is the greek word for what's left over after you take away Something self-similar so all of the recursive subdivided area here is what's left over when you take away the red Which is self-similar Now what's interesting about this is the structure where is the golden bee and all this? Well, you start out with three of them right like that And you're left over with the red and I decided well, I think that's really bold and interesting It's asymmetric and I really like the idea of just having that there left over And recursively subdividing all this black and white and so you can apply this recursive subdivision down to about 14 15 levels and get 40,000 of these things Now the interesting thing is that there's this line across the top Which is the original rectangle so obviously there's it's straight But then you can see that there are other cleavage lines in here that Low and behold are spaced according to you guessed it the golden ratio And it goes horizontally as well as vertically and I was looking at this thing. Oh, those are interesting cleavage lines and then I was looking really close and you notice something kind of funny, which is there's a A break in this cleavage line. There's one golden be tile Right there at this sort of it's it's the locally largest in and it's where diagonals of these these lines intersect and so I said well, that's interesting. So let's count those up. There's one there. Where's the next one's the next line Well, there's one over there There's two here There's three here There's five here There's eight here Land low and behold, there you go. The Fibonacci numbers are back So, you know the Fibonacci numbers and the golden ratio are intimately connected in all sorts of interesting ways And I really I think this is a successful piece of of art as well as mathematical art as well as mathematics And that it lives really up there close to the top of that of that arc Because in some sense, it's you know, it's like the best of most worlds It's it's platonic, but it's and yet it's rich in texture and it looks like it's you know You've made these choices and it's it's very striking visually and it's it's fun to see very when it's very large. Anyway, uh, so Conclusion so recursive to subdivisions are really the trees of linear transformations applied iteratively And that's why you get fractals because there's you know, fractals are generally Generated iteratively and phi is this balance point it keeps keeps these iterations stable and well behaved which has graphical implications and you get these incredible textures and patterns and They're apiatic and so there's this rich rich visual world that that accrues now with respect to the goldenish Rectangle, I just got to say I'm sorry as a self-referential construct the usual miserable goldenish rectangle deserves at least 61.803 percent less respect or worship than it generally Garners it plainly suffers from what I call a golden identity crisis Unlike the true golden rectangle and whereas the the true golden rectangle the one by root five rectangle is a lean gold mean aesthetic fist fighting machine So to them the infidel miserable golden rectangle this aficionado most confidently shouts To phi heaven phi on thee you perfidious figment of fine platonic design A new rectangles in town your days of divinity are finitely numbered and your History even that the peace silent there is finally finished. I for one root for root five by one aman so The speaker being me disavows any support for any tenet of proportional divinity sacred geometry numerology or other Faith-based attempts to import mysterious meaning to the myriad emergent phenomena Of mathematical constraints whose beauties can only be understood from a position of enlightened and rational strength kafiat emptor So this is based on a paper I wrote. Thank you It's on the bridge's website if you want to read it It's uh, it doesn't have stuff about the triangles in it. This talk is an expansion of that I create a g clay prints of various sizes and If you're really really like some of these these designs I actually spent a lot of my time On researching and and playing with space filling curves, which are a different form of recursive geometries And some of the scarves I've designed are available in the shop upstairs and I'm also working on an e-book for the mac and for the ipad called the outside in inside gone, which is Ungeneralized recursive Hilbert and heft domino curves and hopefully that will be out sometime this year. So Quad era demonstrandum or uh questions and generating discussion. Thank you All right. I see a hand up in the back You alluded to earlier about You know the connection with the nautilus shell or perceived connection with the nautilus shell A lot of your patterns that you're putting up there Reminded me of uh snakes and reptile patterns. Yeah, right. Yeah, I guess what I was wondering is in any of the research Explorations is anybody looked at connections between Patterns that occur in nature and some of these. Oh, yeah, absolutely And you know the fibonacci numbers occur in nature But that's because there it's a discrete pattern. And so there's a stability district to discreet, you know pine cones or pineapple or sunflower See sunflower florets and a lot of other flower florets. You can find Spirals that are based on fibonacci numbers where you count them in one counterclockwise Or clockwise you get fibonacci numbers or adjacent fibonacci numbers and and you know, there's anywhere in nature where there's growth There can be constraints and the thing about nature is the best way we have to To model nature to understand nature is to use the math that we know and it turns out it's really good at modeling a lot of things in nature But you never ever quite know whether your mathematical model is really the accurate explanation It might be the best explanation or the simplest mathematical explanation you have But you never quite know whether it's really accurate or not, okay You know, for instance newton law newtons laws of motion They seemed like they were pretty solid for their a couple hundred years and it turns out that they were really kind of approximations That don't always work And so the golden ratio is a little too pristine whereas the fibonacci numbers actually, you know occur and and you know They're they're intimately related But people like to Create meaning And you know so the reason that we have the golden ratio celebrated in books on sacred geometry and so forth is because people like to give meaning to it it makes them feel happy and And designers do that and you know people have done it with a nautilus shell, etc So it's it's it's a little intellectually dicey to to do that It's an interesting exercise to do actual real scientific testing with but nobody does those actual real scientific tests on it um, you know, you could you could test the Preferences of every human on the planet as to which rectangle they like the best And average them all out so that you know the exact answer And what would you do if it turned out to be 1.5733214 instead of the golden ratio? It's close to the golden ratio So maybe there's maybe there's something interesting going on there having to do with people's preferences for asymmetry But it's not clear that there's any Relationship to the golden ratio at all It's just this ideal It happens to be very simple and we have this belief that the world is governed by simple laws So, you know, maybe it'll work out, but it's just a belief Um, has there been any mathematical art research done with the lucas numbers which are related to um They're like a conjugate cousin of the Fibonacci's the lucas numbers I'm not really aware of them. I mean they they grow in the same I mean they have you know adjacent lucas numbers I believe have the same ratio as the golden ratio Like I think because it's the same or it's basically the same recurrence formula But with different initial conditions So, um, you know, ultimately I think it's basically it converges to the same golden ratio And so I wouldn't I There might be really interesting differences down in the low The low numbers because there's only so many low numbers to Fit the patterns we want to use them for but I'm not aware of that Does anything interesting happen if you attempt to Make some similar mapping in non-euclidean space? Probably You haven't tried But but you know the thing is is that There are mappings from euclidean space to non-euclidean space So if something interesting happened there, it's kind of likely maybe I don't know maybe not That you would have an analog back in euclidean space that was interesting too But you know, it's all about exploring it and seeing what happens. All right, let's give another hand for dug