Mathematics in Our World

In this video, we're going to talk about the nature of mathematics, which is mathematics in our world. All of us use mathematics every day. From the time we wake up, we take a look at the clock and count how many hours we have spent sleeping. Right, numbers. Certainly, we use numbers every day. Whenever we do shopping, be it at the mall or online, we spend money and of course, it involves mathematics. Whatever field we are in, surely there is mathematics. In finance, mathematics is used to observe the stock market. Economists who studied the law of demand use mathematics. In astronomy, how could someone measure the path of such extremely gigantic objects just like comets from a very far location? The comet is called NEOWISE. It's expected to make its next pass almost 7,000 years from now. In space missions, mathematics makes space missions accurate in terms of measurements, wherein a slight mistake in mathematics could put a spacecraft out of its target location. Of course, in forecasting weather conditions and climate-related issues that has to be addressed, there is mathematics. Even in forecasting some phenomenal events few hours or days before it occurs, we use mathematics. In sports, whether it's ice skiing, ice skating, basketball, name it, there is mathematics. Even in betting who will win or even guessing the last digits of the final scores in a basketball game. Even Harry Potter's invisibility clock is now being studied to make it real. The possibility of this lies in the novelty of physics and mathematics behind it. In 2006, researchers at Duke University first described something called a metamaterial. The material could capture light coming in from one direction, channel it around the object, and spit it out 180 degrees from the direction it came in. The object in the middle would disappear to the observer. We're able also to observe that in nature there is mathematics. Yes, patterns. In ice crystals, there is fractals, and that is math. This is called a Koch snowflake, and it has a special property. No matter where I look or how much I zoom in, I will see the same pattern over and over. Never-ending patterns like this that on any scale, on any level of zoom, look roughly the same are called fractals. In the arts, do you know that Vincent van Gogh's painting The Starry Night? Well, they discovered that there's a distinct pattern of turbulent fluid structures, which is close to Kolmogorov's equations hidden in many of van Gogh's paintings. Now you know that certainly mathematics is everywhere. Now in this video, we're going to highlight two topics, which are Fibonacci sequence and the golden ratio. Now who would think that there is math in snails? This is what you see when you cut the shell of snails in half. Now this is what we call the golden spiral, which is a series of squares with length of signs 1, 1, 2, 3, 5, 8, and so on. That sequence of numbers is what we call the Fibonacci sequence. The Fibonacci sequence was first introduced in 1202 in a book called Liber Abaci, which is called as the Book of Abacus by an Italian mathematician, Leonardo Pisano Bigoglio, also known as Fibonacci. Well, he came about this sequence of numbers by asking himself, how do rabbits breed? The idea is it takes a month for a young pair of rabbits to become an adult, and it takes another month before an adult rabbit bears another pair of young rabbits. Therefore, initially at month zero, there exists a young pair of rabbits. After one month, the pair of young rabbits become adults. Still, we have one pair of rabbits. After another month, the pair of adult rabbits, of course, they are still adult, but the adult pair of rabbits bear young rabbits, which leads now to two pairs of rabbits. After another month, the pair of adult rabbits bear another pair of rabbits, and the pair of young rabbits become adult, which leads to three pairs of rabbits. After another month, the pair of adult rabbits would still be adult rabbits, but they were able to bear another young pair of rabbits, And the pair of previously young rabbits become adult rabbits. Now we have five pairs of rabbits. And the process still continues wherein the adult pair of rabbits would still be adult, which were able to bear another pair of rabbits. And the previous pair of young rabbits are now adult rabbits, which leads now to eight pairs of rabbits. Now this number of pairs of rabbits are what we call the Fibonacci sequence, wherein we have 1, 1, 2, 3, 5 and 8. On how the Fibonacci sequence is derived, all you have to remember is that the first two terms of the sequence, which is F1 and F2, both have to be 1. Now to get the next number, all you have to do is to add the last two consecutive terms to get the next term, which in this case, if we add 1 and 1, we have 2. To get the next number, all we have to do is to get the last two numbers and then add it to get the next number. So 1 plus 2, we have 3. Then to get the next number, all we have to do is to add the last two numbers. In this case, we have 2 and 3. Add them together, we have 5. To get the next number, all we have to do is to get the last two consecutive numbers, 3 and 5. Add them together, we have 8. Then get the last two numbers again. 5 plus 8, we have 13. 8 plus 13, we have 21. 13 plus 21, we have 34. And the sequence of numbers continues. So all you have to remember to get the nth term of the Fibonacci sequence starting from the 3rd term is to get its last two numbers. So if you want to get, for example, the 10th term of the Fibonacci sequence, all you have to do is to get F sub 10 minus 2 plus F sub 10 minus 1, which is F sub 8 plus F sub 9, which gives us 21 and 34, which is 55. So we know that F sub 10 here is just 55. Now Fibonacci sequence may also be presented in squares, starting with 1 and 1, meaning we have one pair of squares, in which the length of the side are all equal to 1. The next square we have has a side which has a length of 2. The next square, of course, has a side length of 3, followed by a square with a length side of 5, a square with side length of 8, 13, 21, 34, and so on. Fibonacci numbers are said as one of nature's numbering systems. The number of petals, for example, in nature are Fibonacci numbers. Flowers like white calla lily have one petal. Euphorbia has two petals. Trillium has three petals. Hibiscus have five petals. Bloodroots have eight petals. Black-eyed Susan have 13 petals. Shasta daily have 21 petals. DC have 34. Now know that not all flowers have number of petals, which is a Fibonacci number. It is that most flowers have number of petals, which is a Fibonacci number. Now you know about the Fibonacci sequence, it's time to listen to a music which is from Fibonacci numbers. Now moving on, let's have the golden ratio, which is symbolized by 5, which is approximately equal to 1.618. Using the Fibonacci sequence, we may get the value of the golden ratio by getting the ratio of two consecutive numbers in the Fibonacci sequence, starting with 1 and 1, 2 and 1, 3 and 2, 5 and 3, 8 and 5, 13 and 8, 21 and 13, 34 and 21, and so on. Now the value of 1 divided by 1, of course, is 1.0. 2 when divided by 1 is 2.0. 3 divided by 2 is 1.5. 5 divided by 3 is approximately 1.67. 8 divided by 5 is 1.60. 13 divided by 8 is 1.625. 21 divided by 13 is approximately 1.615. 34 divided by 21 is approximately 1.619. Now observing the values, we have 1, 2, 1.5, 1.67, 1.60, 1.625, 1.615, 1.619. Now what you must notice is that these ratios or quotients get closer and closer to the value of the golden ratio, which is approximately 1.618. And the larger the pair of consecutive Fibonacci numbers that you choose, the more close you are to the value of the golden ratio. Now how do we compute for the exact value of the golden ratio? The concept is you have a line segment, which in this case is represented by this line segment, which when divided into two sections, we have a longer segment and a shorter segment. We call the longer segment as B, and we call the shorter segment as S. Now the ratio between the longer segment and the shorter segment, the ratio between this segment and this segment, which is represented by B over S, should be equal to the ratio between the entire line segment, the entire line segment, which is S plus B, and its longer segment, which is B. Applying this, we may get the golden ratio, which is exactly equal to 1 plus square root of 5 all over 2, which is approximately 1.6188. Now let's compute for that exact value. Remember, we have B over S, which is equal to S plus B over B. Now to get the exact value of the golden ratio, we may set the length of a segment to 1. In this case, we're going to let this value of S be equal to 1. So meaning, these values of S are set to value of 1, which gives us B over 1, which is equal to 1 plus B over B. Now by cross multiplication, we may transfer B on the left side, and 1 on the other side. So that will give us B squared, which is equal to 1 plus B. Transferring this 2 on the left side, we're going to have B squared minus B minus 1, which is equal to 0. Now what you must notice is that this equation is quadratic. And how do we solve quadratic equation if we have AX squared plus BX plus C is equal to 0? We're going to make use of the quadratic formula, in which in this case, the value of small a is equal to 1, the value of small b is equal to negative 1, and the value of small c is negative 1. And to get the value of the variable, we're interested in this case, we have the capital B. All we have to do is to substitute negative B, which is negative of negative 1, plus minus square root of B squared minus 4AC over 2A, which gives us negative of negative 1, so that is 1 plus minus square root of positive 1 minus negative 4, so that is 5, over 2 times 1, we have 2. And of course, since B here refers to this length, which cannot be negative, so we're going to ignore 1 minus square root of 5. So that gives us the value of B as 1 plus square root of 5 over 2. So B over S, which is B over S is 1, is still 1 plus square root of 5 over 2, which is approximately 1.618. Now let's verify if that is equal to S plus B over B. So we have S plus B over B, so that is 1 plus B, which is approximately 1.618, divided by B again, which is 1.618, and we're going to have 2.618 divided by 1.618, which is surprisingly would still be 1.618. So clearly, this shows that B over S is equal to S plus B over B. Now alternatively, there's another way to get the exact value of the golden ratio using B over S is equal to S plus B over B. Now since B over S is equal to S plus B all over B, then we say that B over S is equal to S over B plus B over B, which is equal to 1 over B over S, the reciprocal of this, and then B over B is equal to 1. Now assigning a variable, let's say phi, as B over S. This gives us as B over S, which is phi, is equal to 1 over B over S, or in other words, 1 over phi plus 1, which we may rewrite as 1 plus 1 over phi. Now since phi, given this, is 1 over phi plus 1, then we may replace this phi as 1 over phi plus 1. So meaning phi is equal to 1 plus 1 over phi, phi, which is 1 plus 1 over phi. Now remember that phi is 1 plus 1 over phi. So phi is equal to 1 plus 1 over 1 plus 1 over this phi, remember, is 1 plus 1 over phi. Then continuing this, we may substitute again this phi as another 1 plus 1 over phi, and it will not end. So it follows that phi is just a continuous nested fraction, which is 1 plus 1 over 1 plus 1 over 1 plus 1 over 1 plus 1, and so on. Now how do we really get the golden ratio? Since phi here is 1 over phi plus 1, and by multiplying both sides of the equation with phi, this will be phi squared, which is equal to 1 plus phi. And transferring this 2 on the other side gives us phi squared minus phi minus 1 is equal to 0, which is the similar quadratic equation that we encountered on the previous slide. So getting the value of phi here would surely be equal to 1 plus square root of 5 over 2. So this shows that the value of phi, which is b over s, is equal to 1 plus square root of 5 over 2. The golden ratio has been observed in many works of art, like the Parthenon of Greece, the Great Pyramids of Giza in Egypt, and even Leonardo da Vinci's Mona Lisa and the Vitruvian Man. The Vitruvian Man, also known as the Universal Man, represents beauty, complexity, and symmetry of the human frame. The proportions of man, the paintings of Leonardo da Vinci, has been appreciated and raved about for a very long time. One of his lesser known but highly acclaimed work is the Vitruvian Man, based on the work of the architect Vitruvius. Vitruvius had shown a man in two superimposed positions, with his arms and legs apart, and inscribed in a circle and square. The drawings are often called the Canon of Proportions. The drawing is in pen and ink and kept at an art gallery in Venice and opened to the public only occasionally. The drawing is based on the correlations of the ideal human body in relation to geometry. Vitruvius described the human body as being the ideal body to determine correct proportions in architecture. Vitruvius determined that the human body should be eight heads high. Using Roman architect Vitruvius, Leonardo da Vinci claimed that the navel is center of the human body, such that when a center is drawn around, it encompasses both stretched hands and legs. Also, a square can be drawn taking crotch at the center as the height of a standstill body is equal to total arm span length. A palm is four fingers. A foot is four palms. A cubit is six palms. Four cubits make a man. A pace is four cubits. A man is 24 palms. If you open your legs enough that your head is lowered by 1 by 14th of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the center of the extended limbs will be the navel and the space between the legs will be an equilateral triangle. These proportions are also given by him. The length of the outspread arms is equal to the height of a man. From the hairline to the bottom of the chin is 1 by 10th of the height of a man. From below the chin to the top of the head is 1 by 8 of the height of a man. From above the chest to the top of the head is 1 by 6 of the height of a man. From above the chest to the hairline is 1 by 7 of the height of a man. The maximum width of the shoulders is 1 by 4 of the height of a man. From the breasts to the top of the head is 1 by 4 of the height of a man. The distance from the elbow to the tip of the hand is 1 by 4 of the height of a man. The distance from the elbow to the armpit is 1 by 8 of the height of a man. The length of a hand is 1 by 10 of the height of a man. The root of the penis is at half the height of a man. The foot is 1 by 7 of the height of a man. From below the foot to below the knee is 1 by 4 of height of a man. From below the knee to the root of the penis is 1 by 4 of the height of a man. The distances from below the chin to the nose and the eyebrows and the hairline are equal to the ears and to 1 by 3 of the face. Now, do you know that even our face possess golden ratio? Take a look at this. There are several golden ratios in the human face. For example, the total width of the two front teeth in the upper jaw over their height gives a golden ratio. The width of the first tooth from the center to the second tooth also yields a golden ratio. These are the ideal proportions that a dentist may consider. Some other golden ratios in the human face are the length of face over the width of face, the distance between the lips and where the eyebrows meet over the length of nose, the length of face over the distance between tip of jaw and where the eyebrows meet, the length of mouth over the width of nose, the width of nose over the distance between nostrils, the distance between pupils over the distance between eyebrows. Although many claim that mathematicians are just forcing the golden ratio in the painting of Mona Lisa, still, the golden ratio seems to be observed in that painting. As Leonardo da Vinci said, which translates to, let no one read me who is not a mathematician. Now, it's time for you to take a look around and observe because the next pattern might be named after you. Speaking of pattern, let's see how good you are in identifying patterns. Given these five figures, can you identify which is the odd one out? The answer is D. It's simply because if you have this figure, and you try to rotate it clockwise, rotating it slowly, first you're going to have B, rotating it again, you're going to have E, rotating it again, you're going to have C, and rotating it again, you're going to have A. So, no matter how you rotate it clockwise, you will never have the same figure as D. So, D is the odd one out. Another one, which of the following figures is the odd one out? The answer is B. So, just like the previous example, try to rotate any of A, C, D, and E, rotate it clockwise, and you'll get A, C, D, and E. Now, this ends our topic, Mathematics in our World. If you happen to like this video, don't forget to hit like, subscribe, and hit the bell button so that you'll be updated in my next video. We would like to thank Professor Milo Ditesha and Mr. Eugenio Cedric Coro for contributing to the content of this presentation.