The absurd golden ratio Robb Enzmann TEDxMiamiUniversity

good afternoon everyone contrary to the title of my presentation I'm not really here to talk about the golden ratio I'm here to convince you that mathematics isn't scary because most people tend to end their relationship with math the way they end up the rest of their high school relationships which went something along the lines of we're finished and I never want to see you ever again so to convince you that there is a beautiful side to math I'm going to use the golden ratio as my discussion point so to begin we have to be good mathematicians and actually construct it and I'm going to do that using the world-famous Fibonacci numbers so some of you may have heard of and here's how we're going to make them I'm going to start with just a square let there be a square which has a side of length 1 then I can use that square to build another square on the side it would have the length 1 as well then along the bottom it would have a length of 1 plus 1 which is 2 and then you can do 2 plus 1 which is 3 3 + 2 is 5 and the pattern unfolds you can even do this at home and discover some more Fibonacci numbers and you can keep doing this as long as you like until you end up with something that looks like this what we often call a golden rectangle or a golden spiral inside of it and we call it golden because of a really interesting property that it has which is what the funky symbols at the bottom are saying it says if you take the length of the top and you divide it by the length of the side it's always going to be about 1.618 ish and this was proved in the late 18th century by an astronomer called Johannes Kepler and here's kind of what Kepler's proof looked like he just walked down the Fibonacci sequence and started dividing numbers by the ones that came before it and you're going to start seeing a pattern as I go down you'll see that the one point six isn't changing and now if I keep going then one point six one isn't changing and this pattern will continue the further I go down the sequence dividing these numbers the closer I get to this number which is one point six one eight dot dot dot and you may have encountered the dot dot dot before when you learned about irrational numbers think of pi pi was that irrational number that everybody had to learn about i'm going to call this one phi or the golden ratio because that's easy to say and a lot shorter than writing an infinite number of digits when these funky irrational numbers were first translated out of Latin they were called surd numbers which the English bastardized into absurd numbers because they are kind of absurd they just go on forever without repeating but perhaps more absurd than that is what's happened to the golden ratio recently it has suffered because of the media just like every celebrity you do one google search for the golden ratio and you're going to see a picture like this a Parthenon a Mona Lisa a Sistine Chapel or a person's body with the golden ratio stuck on top of it this is not how the golden ratio works this is putting a camera somewhere so that it fits and aesthetics are a lot more complicated than this there isn't one magical number that determines what we do and do not like to look at I can give you a very quick example to explain this this is an example of an experiment that was done in the nineteenth century where they people were just handed a group of rectangles and I would love it if you would just pick your favorite one right now out of it and it can be any of these rectangles and if you're like most people you picked the bottom right one and not this one which is the only golden rectangle up there I like the tall skinny one because it kind of looks like me so it's there's more to it than simply just one number that works but this isn't to say however that there are no civilizations that have used the golden ratio we know for a fact that the ancient Mayans used it and this isn't because we drew lines on top of their doors it's because we went and we looked at the doors which were made of stone and there were Nick marks and actual carvings in the door that indicated they used a geometric construction of the golden ratio much like how we built it with those squares earlier but obviously the Mayans did not have the level of mathematics we have today they didn't use the same rigor but they did what was important they looked for patterns around them specifically patterns and things like these this is a sunflower and you can see the spirals coming from the center and going clockwise and counterclockwise and it turns out these are the golden spirals that we're looking for and it doesn't seem very intuitive at first but the Mayans saw these patterns they looked at the petals they saw the petals growing these spirals and they were usually a Fibonacci number of spirals how could this possibly be well here's a thought experiment to try and see how this works imagine that you're a sunflower and you want to pack as many seeds as possible onto your disk because then you know more likely to pass on your seeds have lots of children and Darwin smiles on your existence so as you're growing out you have to figure out where you're going to place each of the seeds right well let's say as you're growing out you decide to put one every 1/3 of the circle does for whatever reason and then you'll notice that these just get these straight lines of seeds they all stack on top of each other this is not good this is a lot of empty space that you could have had seeds so you're less likely to live very long as a sunflower that uses 1/3 but if instead you use one fie you go a5 the way around the circle every time they're never going to overlap and this has to do with the fact that Phi is so far from a simple ratio but here's a catch we know when we were dividing those Fibonacci numbers earlier that it got really close to Phi so what actually happens is you get a Fibonacci number of arms that are all packed together as closely as possible and this you can very clearly count them there's 21 going that direction and then 13 going the other way and of course 13 and 21 are both Fibonacci numbers that are right next to each other and as a botanical example this is an aloe plant where you can clearly see those five spirals going out but it also brings up an interesting point about the number five in the golden ratios history I want you to imagine that you're tiling your bathroom floor but you want to do something really cool you want to be able to pick it up turn it 1/5 of the way around put it back down have it look exactly the same but K is this Pentagon because you know your bathroom floor is a rectangle and you can't use a bunch of Pentagon's because it has these holes in the middle of it so if you turn it it's not going to fit in the 1970s Roger Penrose did something extremely clever he took this Pentagon and a pentagram inside of it and then he saw if you have the length of the side to be 1 and the diagonal will always be Phi then you took this triangle and this triangle and made this tiling pattern which is one beautiful and it has that really interesting property where if you turn it and rotate it 1/5 of the way put it back down it will fit perfectly on top of itself this shattered our conception of rotational symmetry it was thought to be impossible until the 1970s was extremely recent and you don't just have to take my word for it that these things exist you can go visit some here at Miami University this is bachelor halls courtyard so the takeaway from all of this is that you don't have to be a mathematician to enjoy math and employ it in your life you just simply need to look for patterns around you and then challenge yourself have I avoided the confirmation bias can I logically deduce what I'm seeing here and if you can answer these questions in the affirmative then perhaps you've become the genesis of an idea that changes the way all of us think thank you very much

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