[Music] hello everyone my name is ram and welcome to another video of matu klassan some people may think that mathematics was made to torture our brains on nonsense numbers but what if math is already present in nature and mathematicians are just translating it to us using symbols like numbers have you ever counted the number of petals in a flower you might think that any number is possible but you might be surprised because nature seems to favor a particular set of numbers like 1 2 3 5 8 and 13. it may seem a coincidence to you but these some sort of numbers form a pattern in the sequence the next number is found by adding up the two numbers before it for example 2 is found by adding the numbers before it one plus one equals two following the same pattern one plus two equals three two plus three equals five three plus five equals eight and five plus eight equals thirteen this pattern is called fibonacci sequence early writing show that this pattern of numbers was already investigated by some indian mathematicians like pingala virahanka and hema chandra though this pattern was popularized in europe by leonardo of pisa also known as leonardo fibonacci thus the name of the pattern he's one of the most influential mathematicians of the middle ages because hindu arabic numeral system which we also use today was popularized in the western world because of his book liber abachi or book of calculation in this book fibonacci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions the solution for this problem well as you guessed the fibonacci sequence now let's investigate the problem further if two newborn rabbits are put in a pen how many rabbits will be in the pan after a year assume that rabbits always produced one male and one female offspring can reproduce once every month can reproduce once they are one month old and they never die we start with the first pair of newborn rabbits at the beginning of the month since they are too young to produce an offspring when they get old at the beginning of the second month we still have one pair this time the rabbits are old enough to reproduce so at the start of the third month we will have two pairs on the fourth month the old pair will produce another pair while the first set of offspring will be old enough to reproduce so on the fifth month we will have five pairs of rabbits by using the fibonacci sequence we will have 144 pairs in the beginning of the 12th month however we need to identify the number of rabbits after a year so we need to continue the sequence up to the 13th term of the sequence so the final answer to the problem is 233 pairs or 466 rabbits fibonacci sequence is a wonderful series of numbers that could either start with 0 or 1. let's visualize these numbers using square tiles i will start with a one by one square then another together they form a one by two a rectangle above that a two by two square next to that is a three by three square beneath that a five by five square now if we continue to do this and connect opposing diagonals continuously it will reveal the fibonacci spiral and this spiral could be seen a lot in nature architecture arts human body and beyond going back to the rectangle what if we're going to divide the two dimensions 8 and 13 on this case notice that it's just like dividing two consecutive fibonacci numbers right doing this up to the highest possible pair of fibonacci numbers will give us the golden ratio the golden ratio is approximately equal to 1.618 represented by the greek letter p or phi in math the golden ratio is a term used to describe the relationship of two figures where the numbers seem to be in some form of complementary ratio if you have a number a and a lower number b then the two are in the golden ratio if the quotient of these two numbers somehow a near 1.618 so basically any ratio which comes close to this value is said to be perfectly proportioned the golden ratio is commonly used to calculate perfect spirals triangles arcs and other figures for example a rectangle with a length of 8.09 inches and a width of 5 inches is in the golden ratio because 8.09 divided by 5 is equal to 1.618 since the discovery of the golden ratio many known individuals were inspired to incorporate this magnificent number to their greatest work and creation during the renaissance leonardo da vinci used the proportions set forth by the golden ratio to construct his masterpieces sandro botticelli michael michelangelo and others appear to have employed this same technique in their artwork in a music evidence were found on the work of a debussy mozart beethoven bach and chauffen to mans a marvelous architecture like parthenon taj mahal roman arches egyptian pyramids eiffel tower notre dame cathedral and many more were also built based on this mathematical pattern even in the bible in exodus chapter 25 verse 10 a god commands moses to build the ark of the covenant in which to hold his covenant with the israelites and even the ark which god commands nova to build is based on a variation of the same ratio in the 1490s leonardo da vinci illustrated a mathematical textbook entitled divine proportion by luca patioli in this book along with his work the vitruvian man he pointed out the existence of the golden ratio in the human body this can be proven by using the actual comparisons of the measurements of our bone structures and body it even applies to our dna yep shockingly true psychologists have also noted that many celebrities and other attractive personalities have facial body and bone structures that conform to the golden ratio draenoraya in nature aside from the branching patterns of trees spirals on snails animal horns and seed heads did you know that honeybee colonies maintain a ratio of males to females at 1 to 1.618 this number even extends to natural occurrences earth moon relationship solar system and galaxies you may even have it on your phone yes because aside from the rule of thirds in photography fibonacci spiral is also being used you might want to check that out it's also available in app store and google play so how is this even possible probably some of you will say that they have no practical use but the fibonacci sequence and the golden ratio play a vital role in the finance sector worldwide like helping financial analysts predict and understand the market maybe those are just coincidences and don't mean anything to few however coincidence or not these patterns became part of the world we are living today and they help us unravel the mysteries of nature and before i forget the background music i'm playing it also came from the fibonacci sequence [Music]