Fibonacci Numbers and the Golden Ratio

1. Fibonacci Numbers and the Golden Ratio Say What?

2. What is the Golden Ratio? Well, before we answer that question let's examine an interesting sequence (or list) of numbers. Actually the series starts with 0, 1 but to make it easier we’ll just start with: 1, 1

3. To get the next number we add the previous two numbers together. So now our sequence becomes 1, 1, 2. The next number will be 3. What do you think the next number in the sequence will be? Remember, we add the previous two numbers to get the next. So the next number should be 2+3, or 5. Here is what our sequence should look like if we continue on in this fashion for a while: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…

4. Now, I know what you might be thinking: "What does this have to do with the Golden Ratio? This sequence of numbers was first “discovered” by a man named Leonardo Fibonacci, and hence is known as Fibonacci's sequence. Really Famous Really Smart Math GEEK Leonardo Fibonacci

5. The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Let's look at some of the ratios of these numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…

6. 2/1 = 2.03/2 = 1.55/3 = 1.678/5 = 1.613/8 = 1.62521/13 = 1.61534/21 = 1.61955/34 = 1.61889/55 = 1.618 Since a Ratio is basically a fraction (or a division problem) we will find the ratios of these numbers by dividing the larger number by the smaller number that fall consecutively in the series. So, what is the ratio of the 2nd and 3rd numbers? Well, 2 is the 3rd number divided by the 2nd number which is 1 2 divided by 1 = 2 And the ratios continue like this….

7. Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number (from both sides of the number)! 2/1 = 2.0 (bigger) 3/2 = 1.5 (smaller) 5/3 = 1.67(bigger) 8/5 = 1.6(smaller) 13/8 = 1.625 (bigger)21/13 = 1.615 (smaller)34/21 = 1.619 (bigger)55/34 = 1.618(smaller)89/55 = 1.618 Fibonacci Number calculator

8. 5/3 = 1.678/5 = 1.613/8 = 1.62521/13 = 1.61534/21 = 1.61955/34 = 1.61889/55 = 1.618Notice that I have rounded my ratios to the third decimal place. If we examine 55/34 and 89/55 more closely, we will see that their decimal values are actually not the same. But what do you think will happen if we continue to look at the ratios as the numbers in the sequence get larger and larger? That's right: the ratio will eventually become the same number, and that number is the Golden Ratio!

10. The Golden Ratio is what we call an irrational number: it has an infinite number of decimal places and it never repeats itself! Generally, we round the Golden Ratio to 1.618.

11. Here is the decimal value of Phi to 2000 places grouped in blocks of 5 decimal digits. The value of phi is the same but begins with 0·6.. instead of 1·6.. . Read this as ordinary text, in lines across, so Phi is 1·61803398874...)Dps: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 50 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 100 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 200 06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 500 13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 1000 10767 38937 64556 06060 59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 21221 24154 44006 47034 05657 34797 66397 23949 49946 58457 88730 39623 09037 50339 93856 21024 23690 25138 68041 45779 95698 12244 57471 78034 17312 64532 20416 39723 21340 44449 48730 23154 17676 89375 21030 68737 88034 41700 93954 40962 79558 98678 72320 95124 26893 55730 97045 09595 68440 17555 19881 92180 20640 52905 51893 49475 92600 73485 22821 01088 19464 45442 22318 89131 92946 89622 00230 14437 70269 92300 78030 85261 18075 45192 88770 50210 96842 49362 71359 25187 60777 88466 58361 50238 91349 33331 22310 53392 32136 24319 26372 89106 70503 39928 22652 63556 20902 97986 42472 75977 25655 08615 48754 35748 26471 81414 51270 00602 38901 62077 73224 49943 53088 99909 50168 03281 12194 32048 19643 87675 86331 47985 71911 39781 53978 07476 15077 22117 50826 94586 39320 45652 09896 98555 67814 10696 83728 84058 74610 33781 05444 39094 36835 83581 38113 11689 93855 57697 54841 49144 53415 09129 54070 05019 47754 86163 07542 26417 29394 68036 73198 05861 83391 83285 99130 39607 20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940 98864 00764 43617 09334 17270 91914 33650 13715 2000

12. We work with another important irrational number in Geometry: pi, which is approximately 3.14. Since we don't want to make the Golden Ratio feel left out, we will give it its own Greek letter: phi. Φ φ Phi

13. which is equal to:

14. One more interesting thing about Phi is its reciprocal. If you take the ratio of any number in the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will approach the approximation 0.618. This is the reciprocal of Phi: 1 / 1.618 = 0.618. It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same. In fact, I cannot name another number that has this property! This only adds to the mystique of the Golden Ratio and leads us to ask: What makes it so special?

15. The Golden Ratio is not just some number that math teachers think is cool. The interesting thing is that it keeps popping up in strange places - places that we may not ordinarily have thought to look for it. It is important to note that Fibonacci did not "invent" the Golden Ratio; he just discovered one instance of where it appeared naturally. In fact civilizations as far back and as far apart as the Ancient Egyptians, the Mayans, as well as the Greeks discovered the Golden Ratio and incorporated it into their own art, architecture, and designs. They discovered that the Golden Ratio seems to be Nature's perfect number. For some reason, it just seems to appeal to our natural instincts. The most basic example is in rectangular objects.

16. Look at the following rectangles:Now ask yourself, which of them seems to be the most naturally attractive rectangle? If you said the first one, then you are probably the type of person who likes everything to be symmetrical. Most people tend to think that the third rectangle is the most appealing.

17. If you were to measure each rectangle's length and width, and compare the ratio of length to width for each rectangle you would see the following: Rectangle one: Ratio 1:1Rectangle two: Ratio 2:1Rectangle Three: Ratio 1.618:1

18. Have you figured out why the third rectangle is the most appealing? That's right - because the ratio of its length to its width is the Golden Ratio! For centuries, designers of art and architecture have recognized the significance of the Golden Ratio in their work.

19. Let's see if we can discover where the Golden Ratio appears in everyday objects. Use your measuring tool to compare the length and the width of rectangular objects in the classroom or in your house (depending on where you are right now). Try to choose objects that are meant to be visually appealing.

20. Were you surprised to find the Golden Ratio in so many places? It's hard to believe that we have taken it for granted for so long, isn't it?

21. How about in music? Lets take a look at the piano keyboard…do you see Anything familiar?

22. Count the number of keys (notes) in each of the brackets…You will see the numbers 2,3,5,8,13….coincidence?Does it look like the Fibonacci sequence…it should because it is!

23. How about Architecture?

24. Find the Golden Ratio in the Parthenon. 1. Let's start by drawing a rectangle around the Parthenon, from the left most pillar to the right and from the base of the pillars to the highest point.2. Measure the length and the width of this rectangle. Now find the ratio of the length to the width. Is the number fairly close to the Golden Ratio?3. Now look above the pillars. You should notice some rectangles on the face of the Parthenon. Find the ratio of the length to the width of one of these rectangles. Notice anything?There are many other places where the Golden Ratio appears in the Parthenon, all of which we cannot see because we only have a frontal view of the structure. The building is built on a rectangular plot of land which happens to be ... you guessed it - a Golden Rectangle!

27. Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure. Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.

29. The Golden Ratio in ArtNow let's go back and try to discover the Golden Ratio in art. We will concentrate on the works of Leonardo da Vinci, as he was not only a great artist but also a genius when it came to mathematics and invention.

31. The Annunciation - Using the left side of the painting as a side, create a square on the left of the painting by inserting a vertical line. Notice that you have created a square and a rectangle. The rectangle turns out to be a Golden Rectangle, of course. Also, draw in a horizontal line that is 61.8% of the way down the painting (.618 - the inverse of the Golden Ratio). Draw another line that is 61.8% of the way up the painting. Try again with vertical lines that are 61.8% of the way across both from left to right and from right to left. You should now have four lines drawn across the painting. Notice that these lines intersect important parts of the painting, such as the angel, the woman, etc. Coincidence? I think not!

33. The Mona Lisa - Measure the length and the width of the painting itself. The ratio is, of course, Golden. Draw a rectangle around Mona's face (from the top of the forehead to the base of the chin, and from left cheek to right cheek) and notice that this, too, is a Golden rectangle.

34. Leonardo da Vinci's talent as an artist may well have been outweighed by his talents as a mathematician. He incorporated geometry into many of his paintings, with the Golden Ratio being just one of his many mathematical tools. Why do you think he used it so much? Experts agree that he probably thought that Golden measurements made his paintings more attractive. Maybe he was just a little too obsessed with perfection. However, he was not the only one to use Golden properties in his work.

39. Constructing A Golden Rectangle Isn't it strange that the Golden Ratio came up in such unexpected places? Well let's see if we can find out why. The Greeks were the first to call phi the Golden Ratio. They associated the number with perfection. It seems to be part of human nature or instinct for us to find things that contain the Golden Ratio naturally attractive - such as the "perfect" rectangle. Realizing this, designers have tried to incorporate the Golden Ratio into their designs so as to make them more pleasing to the eye. Doors, notebook paper, textbooks, etc. all seem more attractive if their sides have a ratio close to phi. Now, let's see if we can construct our own "perfect" rectangle.

40. Method One1. We'll start by making a square, any square (just remember that all sides have to have the same length, and all angles have to measure 90 degrees!):2.Now, let's divide the square in half (bisect it). Be sure to use your protractor to divide the base and to form another 90 degree angle:

41. Measure the length of the diagonal and make a note of it. Now, draw in one of the diagonals of one of the rectangles

42. Now extend the base of the square from the midpoint of the base by a distance equal to the length of the diagonal

43. Construct a new line perpendicular to the base at the end of our new line, and then connect to form a rectangle:Measure the length and the width of your rectangle. Now, find the ratio of the length to the width. Are you surprised by the result? The rectangle you have made is called a Golden Rectangle because it is "perfectly" proportional.

44. Constructing a Golden Rectangle - Method TwoNow, let's try a different method that will relate the rectangle to the Fibonacci series we looked at. We'll start with a square. The size does not matter, as long as all sides are congruent. We'll use a small square to conserve space, because we are going to build our golden rectangle around this square. Please note that the golden area is what your rectangle will eventually look like.

45. Now, let's build another, congruent square right next to the first one:Now we have a rectangle with a width 1 and length 2 units. Let's build a square on top of this rectangle, so that the new square will have a side of 2 units:Notice that we have a new rectangle with width 2 and length 3.

46. Let's continue the process, building another square on the right of our rectangle. This square will have a side of 3:Now we have a rectangle of width 3 and length 5.

47. Again, let's build upon this rectangle and construct a square underneath, with a side of 5:

48. The new rectangle has a width of 5 and a length of 8. Let's continue to the left with a square with side 8:

49. Have you noticed the pattern yet? The new rectangle has a width of 8 and a length of 13. Let's continue with one final square on top, with a side of 13:

50. Our final rectangle has a width of 13 and a length of 21. Notice that we have constructed our golden rectangle using square that had successive side lengths from the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...)! No wonder our rectangle is golden! Each successive rectangle that we constructed had a width and length that were consecutive terms in the Fibonacci sequence. So if we divide the length by the width, we will arrive at the Golden Ratio! Of course, our rectangle is not "perfectly" golden. We could keep the process going until the sides approximated the ratio better, but for our purposes a length of 21 and a width of 13 are sufficient.