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Fibonacci Sequence http://www.youtube.com/watch?v=eeAGrFBDehE http://www.youtube.com/watch?v=dr20wqKYYtQ The Great Fibonacci Sequence is everywhere, just look and see!!! The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above would be 21+34 = 55 It is that simple! Here is a longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ... Can you figure out the next few numbers? Makes a Spiral When you make squares with those widths, you get a nice spiral: Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on. The Rule The Fibonacci Sequence can be written as a "Rule" First, the terms are numbered from 0 onwards like this: n= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ... So term number 6 is called x6 (which equals 8). Example: the 8th term is the 7th term plus the 6th term: x8 = x7 + x6 So we can write the rule: The Rule is xn = xn-1 + xn-2 where: xn is term number "n" xn-1 is the previous term (n-1) xn-2 is the term before that (n-2) Example: term 9 would be calculated like this: x9 = x9-1 + x9-2 = x8 + x7 = 21 + 13 = 34 Golden Ratio And here is a surprise. If you take any two successive (one after the other)Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034... In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few: A B B/A 2 3 1.5 3 5 1.666666666... 1.6 5 8 8 13 1.625 ... ... ... 144 233 1.618055556... 233 377 1.618025751... Note: this also works if you pick two random whole numbers to begin the sequence, such as 192 and 16 (you would get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...): A B B/A 192 16 16 208 13 208 224 1.07692308... 224 432 1.92857143... 0.08333333... ... ... 7408 11984 ... 1.61771058... 11984 19392 1.61815754... ... ... ... It takes longer to get good values, but it shows you that it is not just the Fibonacci Sequence that can do this! Using The Golden Ratio to Calculate Fibonacci Numbers And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio: The answer always comes out as a whole number, exactly equal to the addition of the previous two terms. Example: When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8. A Pattern Here is the Fibonacci sequence again: n= 0 1 2 3 4 5 6 7 xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... There is an interesting pattern: 8 9 10 11 12 13 14 15 ... Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...) Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...) Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...) And so on (every nth number is a multiple of xn)…it is beauty, isn’t it? Terms Below Zero The sequence can be extended backwards! Like this: n= ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ... xn = ... -8 5 -3 2 -1 1 0 1 1 2 3 5 8 ... (Prove to yourself that adding the previous two terms together still works!) In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this: x−n = (−1)n+1 xn Which says that term "-n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct 1,-1,1,-1,... pattern. History Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before! About Fibonacci The Man His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy. "Fibonacci" was his nickname, which roughly means "Son of Bonacci". As well as being famous for the Fibonacci Sequence, he helped spread through Europe the use of Hindu-Arabic Numerals (like our present number system 0,1,2,3,4,5,6,7,8,9) to replace Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo. Golden Ratio The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618 It appears many times in geometry, art, architecture and other areas. The Idea Behind It If you divide a line into two parts so that: the longer part divided by the smaller part is also equal to the whole length divided by the longer part then you will have the golden ratio. Beauty This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it? Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape. Do you think it is the "most pleasing rectangle"? Maybe you do or don't, that is up to you! Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way. The Actual Value The Golden Ratio is equal to: 1.61803398874989484820... (etc.) The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number. Calculating It You can calculate it yourself by starting with any number and following these steps: A) divide 1 by your number (=1/number) B) add 1 C) that is your new number, start again at A With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with 2 and got this: Number 1/Number Add 1 2 1/2=0.5 0.5+1=1.5 1.5 1/1.5 = 0.666... 0.666... + 1 = 1.666... 1.666... 1/1.666... = 0.6 0.6 + 1 = 1.6 1.6 1/1.6 = 0.625 0.625 + 1 = 1.625 1.625 1/1.625 = 0.6154... 0.6154... + 1 = 1.6154... It is getting closer and closer! But it takes a long time to get even close, however there are better ways and it can be calculated to thousands of decimal places quite quickly. Drawing It Here is one way to draw a rectangle with the Golden Ratio: Draw a square (of size "1") Place a dot half way along one side Draw a line from that point to an opposite corner (it will be √5/2 in length) Turn that line so that it runs along the square's side Then you can extend the square to be a rectangle with the Golden Ratio. The Formula Looking at the rectangle we just drew, you can see that there is a simple formula for it. If one side is 1, the other side will be: The square root of 5 is approximately 2.236068, so The Golden Ratio is approximately (1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it. Fibonacci Sequence There is a special relationship between the Golden Ratio and the Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... (The next number is found by adding up the two numbers before it.) And here is a surprise: if you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio. The Most Irrational ... I believe the Golden Ratio is the most irrational number. Here is why ... One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this: (In numbers: 1.61803... = 1 + 1/1.61803...) That can be expanded into this fraction that goes on for ever (called a"continued fraction"): So, it neatly slips in between simple fractions. Whereas many other irrational numbers are reasonably close to rational numbers (for example Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...) Other Names The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.