Fibonacci Ratio, Golden Ratio, and 8:5
Date: 06/01/97 at 01:20:39 From: Kevin Brown Subject: Fibonacci Sequences\Numbers and Golden Ratio I was asked to measure and record the body lengths in a diagram. I was then asked to put them into a ratio and these are the ratios I got: 8:5 2:1 79:50 40:89 8:5 I was then asked to comment on the ratios. So far three of them are close to the 8:5 ratio (which is the Golden Ratio, isn't it?). Do you think they are all supposed to be the 8:5 ratio? If so, I'll have to measure it again. Also, I am supposed to investigate whether this statement is mathematically true and why: The total number of keys in an octave on the piano, the number of white keys, and the number of black keys are all Fibonacci numbers. I found out that there are 8 white keys per octave and 5 black keys per octave. Are these Fibonacci numbers? Why? I know that 8:5 is the Golden Ratio, but is it a Fibonacci number? Thank you for your time, Kevin
Date: 06/01/97 at 06:02:16 From: Doctor Mitteldorf Subject: Re: Fibonacci Sequences\Numbers and Golden Ratio Dear Kevin, You're thinking very well about this, but there's some information they're not giving you. Also, sometimes questions just aren't asked very clearly - the teachers and also the people who write the textbooks are only human, and often they don't realize what it's like to be on the other side of the question. The ratios of Fibonacci numbers are: 1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.6666... 8/5 = 1.6 13/8 = 1.625 21/13 = 1.6153... 34/21 = 1.6190... If you keep on going for a long time, the ratios settle down. They all get closer and closer to a number without ever quite reaching it. There's a name for this phenomenon when it happens in math - it's called a "limit". You can say that the limiting ratio of Fibonacci numbers as the numbers get higher and higher is the definition of the Golden Ratio. It's not exactly 8/5, but it's close. The exact value of the Golden Ratio is (1+sqrt(5))/2, which is about 1.6180339... I don't know if you've studied algebra yet, or if you know about quadratic equations and the tricks people use for solving them. In case these things are familiar and interesting to you, here's a fact you might investigate: The ratios of Fibonacci numbers get closer and closer to the Golden Ratio, but they never quite get there. You can prove that if they DID ever get there, then they wouldn't change any more, but would stay equal to the Golden Ratio forever after that. Suppose you get to a Fibonacci number called a. The next one is b, and the ratio b/a = (1+sqrt(5))/2. Then the next Fibonacci number would be a+b, so the ratio of (a+b)/b would be what? In other words, if b/a = (1+sqrt(5))/2, what is (a+b)/b? Can you solve that? Can you simplify your answer? It comes out in a pretty cool way. 8/5 and 89/55 and 43/21 are all Fibonacci ratios, but they're not THE golden ratio. I don't know what your teacher was thinking with those examples s/he gave you. Maybe s/he just wanted you to think about all this and then write about it. An octave on the piano is really 5 black notes and 7 white ones. You can make it 5 and 8 if you include the white notes at both ends. But you could equally well make it 6 and 7 by including the black notes at both ends. And in any case, by the time you got to 2 octaves or 3, you wouldn't have a true Fibonacci ratio any more. I think this goes to show that some of the things people say about Fibonacci numbers are just superstition. Same way some great astronomers of 500 years ago used to try to relate the ratios of the orbital diameters of different planets in our solar system to mathematical ratios from geometry. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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