One of the oldest math problem states that: “How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?” The problem was originated from an Italian businessman who wants to calculate the amount of rabbits that he could trade. The businessman’s name was Leonardo Fibonacci of Pisa; he is also mathematician in the Medieval Era. Taking a look at the problem we see that every in the first and second month, 1 pair will be produced. In the third month, 2 pairs will be produced; one of them is from the original pair and one from the pair that was born in the first month. In the fourth month, 3 pairs will be born. And if we keep on going with the number, there will a resultant sequence representing the amount of rabbits born 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... This sequence is named after Leonardo Fibonacci. The rule is that each term of the Fibonacci sequence is determined by adding the previous two terms together (except for the first 2 terms that don’t follow the rule). The sequence is mathematically represented as: F (n) = F (n – 1) + F (n – 2) FIBONOCCI AND THE GOLDEN RATIO When you have a square and add a square of the same size, you form a new rectangle. If you continue adding squares whose sides are the length of the longer side of the rectangle; the longer side will always be a successive Fibonacci number. Eventually the large rectangle formed will look like a Golden Rectangle - the longer you continue, the closer it will be. SIMPLE ENOUGH RIGHT?! Turns out that the Fibonocci sequence is all around us! EVERYDAY ALL THE TIME In architecture, art, astronomy, nature, and even in your own body. ART AND ARCHITECTURE Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio— believing this proportion to be aesthetically pleasing. ART AND ARCHITECTURE Its use started as early as with the Egyptians in the design of the pyramids. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the pyramids of Egypt. ART AND ARCHITECTURE It has also been used in many other well known, and culturally significant buildings: The Greeks knew it as the "dividing a line in the extreme and mean ratio" and used it extensively for beauty and balance in the design of the Parthenon. The main building of the Taj Mahal was designed using the Golden Ratio. This is why it looks so perfect. The rectangles that served as the basic outline for the exterior of the building were all in the Golden Proportion. ART AND ARCHITECTURE It was used it in the design of Notre Dame in Paris, which was built in the 1163 and 1250. Its use continues in modern architecture, as illustrated in the United Nations building: ART AND ARCHITECTURE "The Mona Lisa,” Leonardo DaVinci's most famous painting, is full of Golden Rectangles. If you draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head, you will have a Golden Rectangle. Then, if you draw squares inside this Golden Rectangle you will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. It is believed that Leonardo, as a mathematician himself, purposefully made this painting line up with Golden Rectangles in this fashion in order to further the incorporation of mathematics into art. ART AND ARCHITECTURE We can even HEAR this sequence! In Mozart’s Sonata No.1 in C major, there are 32 measures in the first section, 68 in the section section, for a total of 100 in the first movement! But this isn’t an ancient concept! In Black Star’s “Astronomy (8th light)” the sequence is used in the lyrics of the chorus. A distant reference, but an interesting fact nonetheless! THE HUMAN BODY The Human Body is proportioned according to the Golden Ratio, which is found using Fibonacci Numbers. The Golden Ratio is a special number approximately equal to 1.618. This number can be found by dividing a line into two parts so that the longer part divided by the longer part is equal to the whole length divided by the longer part. http://www.youtube.com/watch?v=P NQk_GJuZQo THE HUMAN BODY The Fibonocci sequence is also seen in DaVinci's famous study of the proportions of man,"The Vetruvian Man" (The Man in Action), is also full of Golden Rectangles. Unlike the Mona Lisa, where all the lines of the Golden Rectangle are assumed by the mathematician, in "The Vetruvian Man", many of the lines of the rectangles are actually drawn into the image, at least in part. There are three distinct sets of Golden Rectangles in this painting: one set for the head area, one for the torso, and one for the legs. NATURE When observing nature, you may not notice the role Fibonacci Numbers play, but with a small education on these numbers, and you may find yourself picking up on certain trends. The most obvious would be the number of petals on a flower, generally you will find that this number is a Fibonacci Number, 3, 5, 8, 13, etc. However, this is just the beginning. The Fibonacci Sequence can also be seen when observing leaves. NATURE If you start at the bottom leaf and count up the stem, when you come to the next leaf that is in line with the initial starting leaf, generally the number will be in line with the Fibonacci Sequence. IN CONCLUSION: MATH AND SCIENCE ARE ALL AROUND US!! It’s easy to think (especially when students take classes for Gen Ed. Credit) that what we learn doesn’t actually apply to us in real, everyday life. BUT IT DOES!