The golden ratio
The golden ratio
What I want to explore in this video is, given some length of string or a line or some line segment right here, b, can I set up an a, so that the ratio of a to b is equal to the ratio of the sum of these two to the longer side? So it's equal to the ratio of a plus b to a. So I want to sit and think about this a little bit. I want to see is can I construct some a that's on this ratio, this perfect ratio that I'm somehow referring to right here, so that the ratio of the longer side to the shorter side is equal to the ratio of the whole thing to the longer side. And let's just assume that we can find a ratio like that. And we'll call it phi. We'll use the Greek letter phi for that ratio over there. So let's see what we can learn about this special ratio phi. Well if phi is equal to a over b, which is equal to a plus b over a, we know that a plus b over a is the same thing as a over a plus b over a. a over a is just 1. And b over a is just the inverse of this statement right over here. So b over a-- this thing right here over here is phi-- so b over a is going to be 1 over phi. This is going to be 1 over phi. So this is interesting. We've now set up a number, which we're going to call this special ratio, phi is equal to 1 plus 1 over phi. Well just that is kind of a neat statement right over there. First of all, you could, if you subtract 1 from both sides of this, you get phi minus 1 is equal to its inverse. That seems to be a pretty neat property of any number that if I just subtract 1 from it, I get its multiplicative inverse. And so that, already that seems kind of intriguing. But then even this statement over here is kind of interesting because we've defined phi in terms of 1 plus 1 over phi. So we can actually think of it this way. We could say that phi is equal to 1 plus 1 over phi. But instead of writing phi, we're saying, wait, phi is just 1 plus 1 over-- instead of saying phi-- I could say, well, that's just 1 plus 1 over and I could just write phi again or I could just keep on going. I could just keep on going like this forever. I could say that's 1 over 1 plus 1 over and just keep on going on and on and on, forever. And this is a recursive definition of a function, or a recursive definition of a variable, where it's defined in terms of itself. But even this seems like a pretty neat property. But we want to get a little bit further into it. We actually eventually want to figure out what phi is. What is the value of phi, this weird number, this weird ratio that we're beginning to explore? So let's see if we can turn it into a quadratic equation that we can solve using fairly traditional methods. And the easiest way to do that is to multiply both sides of this equation by phi. And then you get phi squared-- let me write a little bit different-- phi squared is equal to phi plus 1. phi squared is equal to phi plus 1. And then, actually, I'm going to take a little bit of a side here. But even this is interesting, because then if we take the square root of both sides of this, you get-- let me scroll down a little bit-- you get phi is equal to the square root of-- and I'll just switch the order here-- the square root of 1 plus phi. So once again, we can set up another recursive definition. phi is equal to the square root of 1 plus phi. And I could write phi there, but hey, phi is equal to the square root of 1 plus and I could write phi there, but hey, phi is just equal to the square root of 1 plus, the square root of 1 plus. And we could just keep going on and on like this forever. So even this is neat. The same number that can be expressed this way, the same number where if I just subtract 1 from it, you get its inverse. It can also be expressed in these kind of recursive square roots underneath each other. So this is already starting to get very, very, very intriguing, but let's get back to business. Let's actually solve for this magic number, this magic ratio that we started thinking about. And really from a very simple idea, that the ratio of the longer side to the shorter side is equal to the ratio of the sum of the two to the longer side. So let's just solve this as a traditional quadratic. Let's get everything on the left-hand side. So we're going to subtract phi plus 1 from both sides. And we get phi squared minus phi minus 1 is equal to 0. And we can solve for phi now using the quadratic formula, which we've proven in other videos. You can prove using completing the square. But the quadratic formula you say, negative b. Negative b is the coefficient on this term right here. So let me just write it down, a is equal to 1, that's the coefficient on this term. b is equal to negative 1, that's the coefficient on this term. c is equal to negative 1, that's the coefficient, or it's really the constant term right over there. So the solutions to this, phi-- and we're actually only going to care about the positive solution because we're thinking about a positive-- when we go to our original problem here, we're assuming that these are both positive distances, so we care about a positive value right over here. We get phi is equal to-- do it in orange-- negative b. Well negative negative 1 is 1 plus or minus the square root of b squared. b squared is going to be 1 minus 4ac. a is 1, c is negative 1. So negative 4 times negative 1 is positive 4. So 1 plus 4, all of that over 2a. So a is 1, so all of that over 2. So phi is equal to 1. And once again, we only care about the positive solution here. This is going to be the square root of 5. If you have 1 minus the square root of 5, you're going to get a negative in the numerator. So we only care about the positive solution. 1 plus the square root of 5 over 2. So this seems like a pretty interesting number. Let's actually take a calculator out and see if we can get the first few places of this magic number phi. So let me get my calculator out. Let's just actually evaluate it. And you might recognize that square root of 5 is an irrational number. And so this whole thing is going to be an irrational number, but I'll prove that in another video, which means it never repeats. It goes on and on and on forever. But let's actually evaluate it. So it's 1 plus the square root of 5 divided by 2. So it says 1.6180339. So let me put that aside. Let me write it down. And this is where it starts to get really interesting and mysterious. So this number right over here is 1.618033988 and it just keeps on going on and on and on, keeps on never terminating, never repeating. So that by itself, it's this cool number. It's this ratio that has all of these neat properties, which are pretty crazy anyway that you express it. But what's really neat is if we revisit this thing right over here. Because what is 1 over phi going to be? So 1 over phi, which we sometimes denote with a capital phi. We already know 1 over phi is just phi minus 1. So we actually can do this in our heads. 1 over this is just going to be 0.618033988. I don't know. There's just something wacky about that, that the inverse of the number is really just the decimals left over after you get rid of the 1. That, by itself, is kind of a crazy idea. But it gets even crazier because this number is showing up everywhere. And as you might imagine from the title of this video, this phi right over here, this is called the golden ratio. This is the golden ratio and it shows up everywhere. It shows up in art. It shows up in music. It shows up in nature. And just to get an idea of where it shows up in nature, it shows up in very pure ideas. So if I were to just draw a perfect star, if I were to just draw a regular star like this. Let me just draw it like this. I'll draw it right over here. So this is just a regular star. All the lengths are equal. So I want to draw it a little bit better than that. So if I just draw a star like this right over here or sometimes this would be called a pentagram, some amazing things start to happen here. The ratio of this pink side to this blue length right over here, that's the golden ratio. The ratio of this magenta to this pink is the golden ratio, as it should, by definition. Now the ratio of the magenta to this orange is also the golden ratio. It just keeps on showing up in a ton of different ways when you look at a pentagram like this. If you look at something like a pentagon, a regular pentagon where all the angles are the same and all the sides are the same, a regular pentagon. If you take any of the diagonals of a regular pentagon, so right over here, if you take this diagonal right over here, the ratio of this green side to--and when I'm talking about the diagonals, ones that actually aren't one of the edges-- the ratio of any of the diagonals to any of the sides is once again, this golden ratio. So it keeps showing up on and on and on. And we can do interesting things with the golden ratio. Let's say that we had a rectangle, where the ratio of the width to the height is the golden ratio. So let's try that out. So let's say that this is its height. This is its width. And that the ratio-- So let's call this a. Let's call this b. And the ratio of a to b is equal to phi. That 1.61 so on and so forth. Let me scroll down a little bit. So that is going to be equal to phi. So that's something interesting to do. Maybe that's a nice looking rectangle of some sort. But let me put out a square here. So let me separate this into a b by b square. So this is a b by b square right over here. And then-- actually let me do it a little bit, let me draw it a little bit differently, this rectangle actually isn't exactly the way I would want to draw it-- so the ratio might look a little bit like this. So the ratio of the width to the length, or the width to the height, is going to be the golden ratio. So a over b is going to be that golden ratio. And let me separate out a little b by b square over here. So this has width b as well. And so this distance right over here is going to be a minus b. So now it is a b by a minus b square. Actually, I should say, we have a b by b square, right over here. This is b by b. And then we're left with a b by a minus b rectangle. Now wouldn't it be cool if this was also the golden ratio? And so let's try it out. Let's find the ratio of b to a minus b. So the ratio of b to a minus b. Well, that's going to be equal to 1 over the ratio of a minus b to b. I just took the reciprocal of this right over here. And this is just going to be equal to 1 over a over b. Let me write this, a over b minus 1. I just rewrote this right there. And that's just going to be equal to 1 over phi. The ratio of a to b, we said, by definition was phi minus 1. But what is phi minus 1? Well phi minus 1 is 1 over phi. It's this cool number. So it's equal to 1 over 1 over 1 over phi, which is once again, just equal to phi. So once again, the ratio of this smaller rectangle, of its height to its width, is once again this golden ratio, this number that keeps showing up. And then we could do the same thing again. We could separate this into an a minus b by a minus b square. Just like that. And then we'll have another golden rectangle, sometimes it's called, right over there. And then we could separate that into a square and another golden rectangle. Then we could separate that into a square and then another golden rectangle. Then another golden rectangle. Actually let me do it like this. This would be better. So let me separate. Let me do the square up here. So this is an a minus b by a minus b square and then we have another golden rectangle right over here. I could put a square right in there. Then we'll have another golden rectangle. Then we could put another square right over there, you have another golden rectangle. I think you see where this is going. Another square, another golden rectangle, which by itself starts to create a cool design that we can keep kind of circling in and in and in. And then, if we actually draw an arc here, something kind of cool happens. If we have an arc that traces these things out, we have something, a pattern that you might have seen many times before. And that pattern does not look too different than what you might see in something like a nautilus shell. And it shows up all over the place in nature. And that makes sense because just the way cells construct themselves. It kind of makes sense to be the same at different scales and the ratio from one scale to the next is maybe the same as the constituent ratios. This right here, and it shows up all over a lot of Leonardo DaVinci's paintings. He never explicitly stated it, but there's a lot of interesting ratios in them. But Salvador Dali, this painting right here, The Sacrament of the Last Supper, he explicitly used the golden ratio. So the actual ratio of the width to the height is the golden ratio, so this is a golden rectangle. And also there's all sorts of ratios and I'll invite you to explore it. The ratio of the different parts of the tables to where it sits in the painting. The golden ratio shows up a ton in this painting. And then he does have the pentagons over here. And we know that the ratio of the diagonal of the pentagon to the sides of the pentagon are also the golden ratio. And so he just thought it was a really cool thing. There's all sorts of neat things. That if you find where these two guys are bowing down are, if you draw that line right over here, this is the golden ratio. The ratio of this length right over here to that length over there. Once again, the golden ratio. It just keeps showing up in this painting. So it's a really, really, really cool thing and I really encourage you to explore this further because it's kind of exciting.