The Golden Ratio: Is It Myth or Math?

a number so perfect perfect we find it everywhere everywhere sacred sacred geometry a mathematical property hardwired into nature secrets the golden ratio the golden ratio the golden ratio what's the answer what's the answer what's the answer history sacred geometry sacred geometric geometry the golden ratio the golden ratio wait wait wait wait hold on i mean really is there actually one special number that underlies everything from sunflowers to seashells everything from pineapples and pine cones to the pyramids and the parthenon i mean a number that can link beauty in art music and the human body one number that links nature's order to the rules of mathematics well some people think so but like uncle carl says extraordinary claims require extraordinary evidence so let's take a closer look at what the golden ratio is really about i mean after all the universe is a strange place full of surprises [Music] hey smart people joe here which of these rectangles is the most perfect give them a look which one just feels most balanced the most beautiful did you pick this one that's a golden rectangle and many people believe that this shape is the most aesthetically pleasing quadrilateral that there is this one not so much this one ew gross yeah get that away the golden rectangle the ratio of its long side to its short side is exactly this this is the golden ratio abbreviated as phi or fee depending on how you prefer to pronounce your greek these numbers after the decimal point they go on forever without repeating like the better known pie phi is an irrational number it's irrational because it can't be written as the ratio of two integers five that's rational because we can write it as the integer five over the integer one the number zero point five also rational we can write it as 3 over 4. even 0.3333 infinitely repeating that's rational because it can be written simply as 1 over 3. but what about the diagonal of a square whose sides are one unit long the pythagorean theorem tells us that the diagonal has a length of the square root of two which is a number but one that can't be written as a ratio of two nice and tidy integers it's irrational likewise phi also can't be written as a simple integer ratio and an ancient greek named euclid was one of the first to notice that this euclid guy was big into geometry in fact most of the geometry that we learn in school is named after him it can be a pretty big deal to get a whole section of math class named after you so around 300 bc euclid wrote a book called elements a collection of most of what was known about math at the time and until the 20th century it was the best-selling book ever other than the bible you could notice that there was one special way to divide a line where the ratio of the whole to the longer segment was the same as the ratio of the longer segment to the shorter one and that ratio is phi well euclid called it the extreme and mean ratio which sounds like what happens when i make a bad tweet the names phi and golden ratio they didn't show up until almost the 20th century anyway the greeks and mathematicians of that time they didn't think of numbers like we do as these strings of digits from zero to nine to them phi was this ratio just like to them pi wasn't 3.14159 etc pi was just the ratio of a circle's circumference to its diameter this is literally the golden ratio and you can do some weird stuff with it the ratio of the long sides of this triangle to its short side is you guessed it phi this is a golden triangle also called a sublime triangle and the angles of that triangle are 72 72 and 36 degrees now if i divide one of the long sides according to the golden ratio and make a smaller triangle there it's another golden triangle same angles and all and that other triangle we just created the length of these sides to the base is one over phi we call this squatty shape the golden gnomon and if i take one golden triangle and stick two golden gnomons on the side i get a regular pentagon yeah we're just getting started let's overlap two golden omans and add a smaller golden triangle on the side you make a pentagram and going back to our golden rectangle if we put another golden rectangle here another inside that and another and so on and so on and draw a curve through all these shapes we get a shape called the golden spiral if that looks familiar it's probably because you've seen an image like this before on the internet and we will be getting back to that very soon no there's even more strangeness if you multiply phi times itself that's the same as one plus phi take one over phi and that's the same as phi minus one this is a weird number okay fine so what phi is weird there's infinite numbers so some of them are gonna be a little strange what makes phi special is that it shows up in a bunch of really unexpected places that are pretty far off from geometry class or at least people claim to find phi in a lot of unexpected places and this is the really interesting thing about phi because as cool of a number as it is on its own it's achieved this almost mythological status this is like the elon musk of numbers and many people say that because we find it in so many places it can't just be a coincidence it must be a sign of some deeper secret about the universe well where does the real story of phi end and the myth begin well if there's one person responsible for the mythological status of phi it's this guy leonardo of pisa aka fibonacci around the year 1200 fibonacci was responsible for bringing hindu arabic numerals into common use across europe these are the numerals that we use today zero through nine and merchants quickly realized that doing arithmetic with these was way easier than roman numerals which is what everyone in europe was using at the time so to teach people how to use these new numbers which had actually already been in use in asia for like a thousand years fibonacci wrote a math textbook liber abakai which just means the book of calculation this book was full of math problems to teach people how to add and exchange currencies and divide and multiply with these new numbers and tucked inside of chapter 12 was this weird problem about rabbits doing what rabbits are known to do that would end up making fibonacci famous imagine you have a pair of rabbits in a field one male and one female no rabbits die or get eaten starting from the second month she's alive every female reproduces each month making a new pair of rabbits one male one female so how many rabbits will they produce after one year you can pause and take a minute to work it out if you want to but it ends up looking like this you might notice something special about the number of pairs of rabbits each month the number of pairs is equal to the sum of the previous two months and after 12 months you'd have 144 pairs this is the famous fibonacci sequence you can carry it on forever just add the previous two numbers to get the next and so on until the end of the universe or until you get bored the reason that we're talking about the fibonacci sequence in a video about the golden ratio is because as you go on in the fibonacci sequence the ratio between numbers gets closer and closer to phi in fact any sequence of numbers that follows the fibonacci rule adding the two previous to get the next trends to phi like this set the lucas numbers follow the pattern and carry it on and the difference between the terms all trends to phi yeah i know it's weird but fibonacci never made that connection himself a guy named johannes kepler did a few hundred years later the same kepler who figured out the math that explains how planets move pretty smart guy it's after that when the fibonacci sequence and phi got together that the myth really took off and people started to claim these numbers were more than just numbers despite phi's seemingly mystical mathematical origins the truly mind-boggling aspect of phi was its role as a fundamental building block in nature plants animals and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of phi to one phi's ubiquity in nature clearly exceeds coincidence that is how one of the greatest writers in all of history put it and that's really the question isn't it does phi the golden ratio the divine proportion whatever grand name you want to give it really show up everywhere in nature or is it our pattern sensing brains making us think that we see it everywhere like when you notice a license plate from another state and suddenly you start seeing out-of-state license plates more than you used to or think you used to well let's look at some places people claim to see phi the human body it's claimed that the ideal ratio of a person's height to the distance from their navel to their feet is phi i probably don't need to tell you that beauty standards in different cultures vary a lot and people come in way too many shapes and sizes for that to be a rule the great pyramid of giza the parthenon notre dame cathedral in paris the taj mahal a handful of ancient buildings that people claim were built with golden ratio dimensions the thing is for an object that's fairly big like a building or complex like a body there are so many ways to measure it and so many measurements you can take that some are bound to be somewhere around a golden ratio apart from each other i mean this video is a 16 by 9 aspect ratio that's pretty close to 1.6 but it's not phi 16 by 10 would be even closer actually lots of places that people claim to see phi in nature are just plain wrong like the ratio of one turn of a dna helix to its width google that and you'll see results that say it's 34 angstroms high per turn and 21 angstroms wide both fibonacci numbers ooh intriguing unfortunately that's wrong these are dna's actual measurements here's the key thing in any example that you find phi isn't approximately 1.6 give or take it's exactly this if people go measuring things looking for the golden ratio they often measure them in ways that ensure that they find the golden ratio our brains love patterns and once we learn a pattern like the usual arrangement of a mouth and a nose and two eyes to make a face we see that pattern everywhere and that brings us to this a nautilus shell a nautilus is a cool little mollusky thing that swims around with a spiral shell and a face full of spaghetti it's become basically the official mascot of the golden ratio in nature the claim is that if you trace the spiral of this shell each ring is a golden ratio away from the next smallest ring etc but people have gone out and actually measured loads and loads of nautilus shells and they aren't golden spirals the ratios vary quite a bit like snail shells or sheep horns it's an example of what's called a logarithmic spiral i mean it's really cool each turn of the spiral grows by the same proportion because the nautilus grows at the same rate but that proportion isn't phi and that's too bad because logarithmic spirals are cool but no one pays attention to them because of this obsession with phi my point is close is not enough if phi is a fundamental building block in nature we should be able to show that it's more than a coincidence that there's some reason behind it being there which brings us to these not every claim about seeing phi in nature is fake fi does show up in nature in a really interesting way and if you've ever looked closely at a pineapple or a pinecone or a sunflower or an artichoke i don't know who closely studies artichokes but maybe you have all of these plant parts show a special kind of spiral let me show it to you here's a pineapple and if you notice on a pineapple there's these spirals going one direction like this and then we can see a spiral going the other direction the other way around the pineapple let me trace this out and make it a little easier for you to see so a little arts and crafts time here and it's okay to be smart this is going to be fun [Music] there's 13 spirals in that direction well now let's count these spirals in the other direction so we've got eight spirals going in one direction and 13 spirals going in the other direction and if those numbers sound familiar that's because they're both fibonacci numbers and you remember from before that within the fibonacci sequence we find the golden ratio okay that's just one pineapple and maybe that's a big pineapple conspiracy coincidence so let's count something else i don't know if you've ever noticed this but pine cones have these adorable little spirals too let's see if there's any five magic going on in there eight spirals going that direction it's pretty spooky [Music] thirteen spirals going the other direction those are fibonacci numbers too pinecone pineapple maybe these should have been called fine cones and pineapples but i digress we can also find fibonacci numbers in these sunflowers this rose this cauliflower this succulent thing this is fun let's count the spirals on this artichoke too and five spirals going in that direction eight spirals going in that direction five and eight are fibonacci numbers as well and there's this the branch out hold on i came prepared for this thing and this the branch of a monkey puzzle tree yes that is its real name this looks like some sort of medieval weapon for the like a plant night or something we can count the spirals on this thing too very carefully i should be wearing safety goggles for this and eight spirals going in that very dangerous direction come here you 13 spirals going the other direction plants can't do math they can't count so why is there this connection well imagine that you're a plant you eat light so the more sun you can catch with your leaves the better so as a stem or a branch grows up or out where do you put your leaves let's put one leaf right here that looks fine and then let's go say half a turn to where we'd be as far as possible from the first leaf that we laid down we can do our half turn rule again and well wait now we're on top of the first leaf and if we if we continue this well we're not going to be catching maximum rays man let's start over and let's pick a let's pick a new fraction of a turn why not a third so we put our first leaf down here and we can turn a third of a turn right here we turn a third of a turn again for our next leaf right here looking pretty good so far a third of a turn from our third leaf now we're back over the top of our first leaf again so that's not gonna work either maybe one over four so we can start here a quarter turn a quarter turn again another quarter turn but there are plants that actually grow like this but we can quickly see as we continue this pattern we overlap our leaves again this can't be the best strategy out there for catching maximum sun and it turns out that if we use any fraction of a circle with a whole number on the bottom our leaves will eventually overlap a rational number isn't going to work but what if we used an irrational number instead and remember that irrational numbers can't be expressed as simple integer ratios and it turns out that phi might be the most irrational number that there is so let's put our first leaf down and then let's go a fraction of a circle one over phi turns around and remember that one over phi is equal to this so if we express that as a fraction of 360 degrees we're taking a turn of about 137.5 degrees each time which probably won't surprise you is called the golden angle that made this little guide that's exactly that angle so let's fill in our leaves using our new phy guide all right so we go 137.5 degrees from our first leaf and we lay another one down 137 and a half degrees from there and lay down our third leaf as we see our new leaves fill the gaps left from the leaves before we never overlap so far let's keep going and see what happens [Music] one two three four five six seven eight the fibonacci number of spirals they form all on their own just from that golden angle turn rule why do these spirals form you can actually do this yourself and play around with different size leaves or petals or whatever and you'll find that with the golden angle as your guide you always put down a fibonacci number of things before you get to these layers where things start to almost overlap but not quite and the spirals just sort of happen from there and there's a fibonacci number of them this works for more than leaves catching sunlight too it's also a useful pattern for catching rain and funneling water down to your roots or for packing more seeds and flower petals into a small space looking at you here sunflowers these are all things that can make you a better plant quick side note we didn't put a camera over there i'll just stay over here quick front note this explains a lot about why plants do this but the how is a lot more complicated and scientists are still working out a lot of the details what we do know is that instead of having some internal leaf angle growth measure thing or something all these angles well they have to do with newly growing plant parts repelling other nearby plant parts kind of like how the poles of magnets if they're alike they repel each other only this is thanks to growth hormones and not magnets but the point is there's actual biology and chemistry underneath all of this so anyway it's not like there's a gene in plants that's programmed to do math or something but plants have had gazillions of years of evolution to find the best way to do all the cool plant stuff that they want to do and it's not like every plant even does this plenty of plants follow different rules and it works just fine for them and that's all that really matters in evolution is that something works well enough not that something reaches some mathematical and irrational golden perfection but i mean you can't deny this is kind of beautiful and i think that's probably a big part of why we're attracted to this pattern more than other plant patterns because you know ape brain like pretty pattern and that's the funny thing about beauty it can take so many forms we know that some artists like salvador dali or the architect le corbusier occasionally used the golden ratio purposely in some of their work but i mean there's plenty of beautiful art that makes no use of the ratio too you remember when i asked you to pick the most beautiful rectangle at the beginning lots of you probably picked one other than the golden rectangle math follows very particular rules and things like beauty and life they're a bit more messy because the world is a messy place and that's part of the beauty isn't it that's okay because sometimes in the middle of that mess if we look hard enough we can find some order after all stay curious okay wow this is this is way creepier than i thought it would be can i be a person again i don't i don't really like being a tropical fruit anyone hello hello you

Channel: It's Okay To Be Smart

Views: 720,433

Rating: 4.9146628 out of 5

Keywords: science, pbs digital studios, pbs, joe hanson, it's okay to be smart, its okay to be smart, it's ok to be smart, its ok to be smart, golden ratio, fibonacci, phi, math, geometry, golden ratio explained, golden ratio in nature, fibonacci sequence, golden spiral, fibonacci numbers, mathematics, golden rectangle, leonardo da vinci, golden ratio spiral, nature, public broadcasting service, itsokaytobesmart, physics

Id: 1Jj-sJ78O6M

Length: 22min 54sec (1374 seconds)

Published: Wed Mar 10 2021

Reddit Comments

[u/NewClayburn]

He should play Hank Green in the biopic.

[u/batatinha_batatinha]

great video

[u/conventionistG]

Three blue one brown does a better job imho.

[u/fantasticdamage_]

Fibonacci !!

[u/willflameboy]

I suppose it's inevitable that people mispronounce phi the same way they mispronounce almost every other Greek letter. That opening sequence is hilarious and worth the video alone.