The Golden Ratio: Is It Myth or Math?

such a perfect number perfect we find it everywhere everywhere sacred sacred geometry a

math

ematical property embedded in the secrets of nature the

golden

ratio

the

golden

ratio

the golden ratio what is the answer what is the answer what is the answer history sacred geometry sacred geometric geometry the golden ratio the golden ratio wait, wait, wait, wait, I mean, is there really a special number that underlies everything from sunflowers to seashells, from pine cones and pineapples to pyramids and the Parthenon? I am referring to a number that can link beauty in art, music and the human body. number that links the order of nature to the rules of

math

ematics well, some people think so, but as uncle carl says, extraordinary claims require extraordinary evidence, so let's take a closer look at what it's really about the golden ratio.

I mean, after all, the universe is a strange place full of surprises hello smart people, joe, here which of these rectangles is the most perfect, give them a look, which one feels the most balanced, the most beautiful, you chose this one that is a golden rectangle and many people believe that this shape is the most aesthetically pleasing quadrilateral. that this one exists, not so much this one, disgusting, yes, take that away from the golden rectangle, the proportion between its long side and its short side is exactly this, this is the golden ratio abbreviated as phi or fee depending on how you prefer to pronounce your in Greek these numbers after the decimal point continue forever without repeating themselves like the best known pie phi is an irrational number it is irrational because it cannot be written as the ratio of two integers five that is 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 repeating infinitely that is 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 it cannot be written as a proportion of two nice, ordered integers, it is irrational in the same way, Phi also cannot be written as a simple integer ratio and an ancient Greek named Euclid was one of the first to notice that this guy Euclid was a big fan of geometry;

In fact, most of the geometry we learn in school is named after him, it can be a big deal to get a whole section of math class named after you, so around 300 BC. C. Euclid wrote a book called Elements, a collection of most of what was known about mathematics at the time and until the 20th century it was the best-selling book after the Bible. Notice that there was a special way of dividing a line where the ratio between the entire segment and the longest segment was the same as the ratio between the longest segment and the shortest segment, and that ratio is phi.

Well, Euclid called it the extreme and mean reason that sounds like what happens when I make a bad tweet. The names phi and golden ratio did not appear until almost the 20th century. Anyway, the Greeks and mathematicians of that time didn't think of numbers the way we do as these strings of digits. from zero to nine for them phi was this proportion just as for them pi was not 3.14159, etc. pi was just the ratio of the circumference of a circle to its diameter, this is literally the golden ratio and you can do some weird things with it, the ratio of the long sides of this triangle to its short side, you guessed it, fi, this It is a golden triangle also called 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 side. proportion and make a smaller triangle there is another golden triangle the same angles and everything and that other triangle that we just created the length of these sides to the base is one over phi we call this squat shape the golden gnomon and if I take a triangle gold and stick two gold gnomons on the side I get a normal pentagon yes we are just starting let's overlap two gold gnomons and add a smaller gold triangle on the side we form a pentagram and go back to our gold rectangle if we put another gold rectangle. here another one inside that and another and so on and so on and we draw a curve through all these shapes, we get a shape called a golden spiral.

If this looks familiar, it's probably because you've seen a picture like this before on the internet and we'll see. I'll get back to that very soon no, there's even more strangeness if you multiply phi by itself, that's the same as one plus phi, take one plus phi and that's the same as phi minus one, this is a weird number, okay, so what phi is weird? infinite numbers, so some of them will be a bit strange. What makes phi special is that it appears in a lot of really unexpected places that are quite far from geometry class or at least people claim to find phi in a lot of unexpected places. and this is what's really interesting about phi because, despite being a cool number on its own, it has reached this almost

myth

ological status.

It's like Elon Musk of numbers and many people say that because we find it in so many places, it may not just be a coincidence, it must be a sign of some deeper secret about the universe. Well, where does Phi's true story end and the

myth

begin? Well, if there is one person responsible for the mythological status of Phi, it is this guy, Leonardo of Pisa, also known as fibonacci around the year 1200 fibonacci was responsible for bringing Hindu Arabic numerals into common use throughout Europe. These are the numbers we use today from zero to nine and merchants quickly realized that doing arithmetic with them was much easier than Roman numerals, which is what everyone in Europe was using at the time to teach children. people how to use these new numbers that had actually already been used in Asia for a thousand years.

Fibonacci wrote a mathematics textbook liber abakai which simply means the calculus book, this book was full of mathematical problems. to teach people how to add and exchange coins and divide and multiply with these new numbers and hidden within chapter 12 was this strange 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, a male and a female, no rabbit dies or is eaten after the second month of life, each female reproduces every month, forming a new pair of rabbits, a male, a female, then, How many rabbits will they produce after one year?

You can pause and take a minute to figure it out if you want, but it ends up looking like this. You may notice something special about the number of rabbit pairs each month. The number of couples is equal to the sum of the two months before and after 12 months. You would have 144 pairs. This is the famous Fibonacci sequence. You can continue it forever, just add the two previous numbers to get the next one and so on until the end of the universe or until you get bored of the reason we are talking about. the fibonacci sequence in a video about the golden ratio is because as you progress through the fibonacci sequence the proportion between numbers gets closer and closer to phi in fact any sequence of numbers that follows the fibonacci rule by adding the previous two to get the following trends to phi so set the luke numbers follow the pattern and continue it and the difference between the terms all the trends to phi yes i know it's strange but fibonacci never made that connection himself some guy named johannes kepler Kepler did the same thing a few hundred years later, who discovered the mathematics that explains how planets move, very smart guy, it was after that, when the Fibonacci sequence and Phi came together, that the myth really took off and people began to claim that these numbers were more than just numbers, despite Phi's seemingly mystical mathematical origins.

A truly mind-blowing aspect of phi was its role as a fundamental building block in nature; plants, animals, and even humans possessed dimensional properties that adhered with eerie exactness to the ratio of phi to one. The ubiquity of phi in nature clearly exceeds coincidence, that's how one of the greatest writers in all of history put it, and that's really the question: isn't phi, the golden ratio, the divine proportion, any great name Whatever you want to give it, does it really appear everywhere in nature or is it our pattern-detecting brains that make us? Think we see it everywhere, like when you notice a license plate from another state and suddenly you start seeing license plates from another state more than you used to see or think you used to well let's look at some places where people say they see phi the human body It is stated that the ideal proportion between a person's height and the distance from the navel to the feet is phi.

I probably don't need to tell you that beauty standards in different cultures vary greatly and people come in 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 they were built with golden ratio dimensions the thing is for an object that is quite large 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 separated by a golden ratio each other.

I mean this video has a 16 by 9 aspect ratio which is pretty close to 1.6 but it's not phi 16 by 10 would be even closer in fact a lot of places where people claim to see phi in the wild are just wrong , like the ratio between one turn of a DNA helix and its width, Google that and you'll see results saying it's 34 angstroms high per turn and 21 angstroms wide, both Fibonacci numbers, oh, intriguing, unfortunately, that's wrong, these are real DNA measurements. Here's the key in any example, that you find that phi is not about 1.6 or so, it's exactly this if people measure things by looking. for the golden ratio, they often measure them in ways that ensure they find the golden ratio.

Our brains love patterns and once we learn a pattern like the usual arrangement of a mouth, nose and two eyes to form a face, we see that pattern everywhere. and that brings us to this a nautilus shell a nautilus is a cool little swimming mollusk with a spiral shell and a face full of spaghetti has basically become the official mascot of the golden ratio in nature the claim is that yes 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 measured lots and lots of nautilus shells and they are not golden spirals, the proportions vary quite a bit, such as 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 in the same proportion because the nautilus grows at the same rate, but that proportion is not phi and that's a shame because spirals Logarithmics are great, but no. one pays attention to them because of this obsession with phi my point is close it is not enough if phi is a fundamental component in nature we should be able to show that it is more than a coincidence that there is some reason behind its existence that brings us For these, no all claims about seeing phi in nature are false.

Fi appears in nature in a really interesting way and if you have ever looked closely at a pineapple, a pineapple, a sunflower or an artichoke, I don't know who up close. study artichokes, but maybe you have all these parts of the plant that show a special type of spiral, let me show you. Here's a pineapple and if you notice that in a pineapple there are spirals going in one direction like this and then we can see a spiral going in the other direction. another direction upside down around the pineapple let me trace this and make it a little easier to see so a little bit of craft time here and it's okay to be smart this is going to be fun there are 13 spirals in that direction okay now let's count these spirals in the other direction, so we have eight spirals that go in one direction and 13 spirals that go in the other direction and if those numbers sound familiar to you it is because they are both Fibonacci numbers and you remember from before that within the Fibonacci sequence we find the golden ratio, okay, it's just a pineapple and maybe it's a big pineapple conspiracy coincidence, so let's tell something else, I don't know if you've ever noticed this, but pineapples also have these adorable little spirals, let's see if there are five magic there eight spirals going in that direction it's pretty spooky thirteen spirals going in the other direction those are fibonacci numbers also pineapple maybe they should have been called thin cones and pine cones but I digress we can also find fibonacci numbersfibonacci in these sunflowers this rose this cauliflower this succulent thing this is fun let's count the spirals of this artichoke too and five spirals going in that direction eight spirals going in that direction five and eight are also Fibonacci numbers and there it is this branch outside wait , I came prepared for this thing and this is the branch of an araucaria yes, that's its real name this looks like some kind of medieval weapon for a plant night or something like that we can count the spirals of this thing very carefully I should wear safety glasses to this and eight spirals going in that very dangerous direction, come here you, 13 spirals going in the other direction, plants can't do math, they can't count, so why is there this connection?

Well, imagine that you are a light eating plant, so the more sun you can catch with your leaves, the better, just as a stem or a branch grows or outside, where do you put the leaves? Let's put a leaf here that looks good and then say half a turn to where we would be. Being as far away as possible from the first sheet that we put down, we can do our half-turn rule again and wait, now we are on top of the first sheet and if we continue like this, we won't be catching the maximum rays, man, let's start again and let's choose a new fraction of return, 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 sheet here it looks pretty good so far, a third of a turn from our third sheet, now we're above our first sheet again, so that won't work either, maybe one over four, so we can start here a fourth . turn a quarter turn again another quarter turn but there are plants that actually grow like this but we can quickly see that as we continue this pattern we overlap our leaves again this may not be the best strategy out there to capture the maximum sun and it turns out that if we use any fraction of a circle with a whole number at the bottom our leaves will eventually overlap, a rational number is not going to work, but what if we used an irrational number instead and remembered that numbers Irrationals cannot be expressed as simple integer ratios and it turns out that phi might be the most irrational number there is, so let's put down our first sheet and then go to a fraction of a circle, one over phi, turn and remember that one over phi is equal to this, so if we express that as a fraction of 360 degrees, we're doing a turn of about 137.5 degrees each time, which probably won't surprise you, it's called the golden angle which made this little guide, that is exactly that angle, so let's fill in our sheets using our new physical guide okay, so let's go 137.5 degrees from our first sheet and place another one 137 and a half degrees from there and place our third sheet while we see that our new leaves fill the remaining spaces of the leaves before we never overlap so far let's go ahead and see what happens one two three four five six seven eight the number of Fibonacci spirals that form on their own just from that golden angle rule of twist why are these spirals formed?

Actually, you can do it yourself and play. with different sized leaves or petals or whatever and you'll find that with the golden angle as a guide you always place a Fibonacci number of things before you get to these layers where things almost start to overlap but not quite and the spirals just sort of overlap. classify They can happen from there and there are a Fibonacci number of them. This works for more than leaves catching sunlight. It is also a useful pattern for catching rain and channeling water to the roots or for packing more seeds and flower petals into a small space.

Here sunflowers, these are all things that can make you a better plant. Quick side note. We didn't put a camera there. I will stay here. Quick front note. This explains a lot about why plants do this, but the how is a lot. It's more complicated and scientists are still working out many of the details, what we do know is that instead of having some measure of internal leaf angle growth or something, all of these angles have to do with parts of plants that grow recently and that repel other parts of nearby plants. about how the poles of the magnets, if they are equal, repel each other only this is thanks to the growth hormones and not the magnets, but the point is that there is real biology and chemistry underneath all this, so of course Anyway it's not like there's a gene in plants that are programmed to do math or anything, but plants have had millions of years of evolution to find the best way to do all the interesting things they want to do and it's not like all plants do this, many plants follow different rules and it works. is fine for them and that's the only thing that really matters in evolution is that something works well enough, not that something reaches some irrational mathematical golden perfection, but I mean, there's no denying that this is a beautiful thing and I think that that's probably a big part of why we're drawn to this pattern more than other plant patterns because you know ape brain is a pretty pattern and that's the funny thing about beauty: it can take so many forms, we know some artists like savior Dali or the architect Le Corbusier occasionally used the golden ratio deliberately in some of their work, but I mean there are many beautiful works of art that also do not use the ratio.

Remember when I asked you to choose the most beautiful rectangle at the beginning? Many of you probably chose something other than the golden rectangle. Mathematics follows very particular rules and things like beauty and life are a little messier because the world is a messy place and that's part of beauty, right? It'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 a lot creepier than I thought. Can I be a person again? No, I don't really like being a tropical fruit.