7.2: The Golden Ratio and Fibonacci Sequence
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In this section, we will discuss a very special number called the Golden Ratio. It is an irrational number, slightly bigger than 1.6, and it has (somewhat surprisingly) had huge significance in the world of science, art and music. It was also discovered that this number has an amazing connection with what is called the Fibonacci Sequence, originally studied in the context of biology centuries ago. This unexpected link among algebra, biology, and the arts suggests the mathematical unity of the world and is sometimes discussed in philosophy as well.
Golden Ratio
With one number \(a\) and another smaller number \(b\), the ratio of the two numbers is found by dividing them. Their ratio is \(a/b\). Another ratio is found by adding the two numbers together \(a+b\) and dividing this by the larger number \(a\). The new ratio is \((a+b)/a\). If these two ratios are equal to the same number, then that number is called the Golden Ratio. The Greek letter \(\varphi\) (phi) is usually used to denote the Golden Ratio.
For example, if \(b = 1\) and \(a / b=\varphi\), then \(a=\varphi\). The second ratio \((a+b)/a\) is then \((\varphi+1) / \varphi\). Because these two ratios are equal, this is true:
\[\varphi=\dfrac{\varphi+1}{\varphi}\nonumber \]
(This equation has two solutions, but only the positive solution is referred to as the Golden Ratio \(\varphi\)).
One way to write this number is
\[\varphi=\dfrac{1+\sqrt{5}}{2} \nonumber \]
\(\sqrt{5}\) is the positive number which, when multiplied by itself, makes \(5: \sqrt{5} \times \sqrt{5}=5\).
The Golden Ratio is an irrational number. If a person tries to write the decimal representation of it, it will never stop and never make a pattern, but it will start this way: 1.6180339887... An interesting thing about this number is that you can subtract 1 from it or divide 1 by it, and the result will be the same.
\[\varphi-1=1.6180339887 \ldots-1=0.6180339887 \nonumber \]
\[1 / \varphi=\frac{1}{1.6180339887}=0.6180339887 \nonumber \]
Golden rectangle
If the length of a rectangle divided by its width is equal to the Golden Ratio, then the rectangle is called a "golden rectangle.” If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because \(a / b=\varphi\). The blue part (B) is a square. The pink part by itself (A) is another golden rectangle because \(b /(a - b)=\varphi\).
Assume that \(\varphi=\dfrac{a}{b}\) , and \(\varphi\) is the positive solution to \(\varphi^{2}-\varphi-1=0\). Then , \(\dfrac{a^{2}}{b^{2}}-\dfrac{a}{b}-\dfrac{b}{b}=0\). Multiply by \(b^{2}, a^{2}-a b-b^{2}=0\). So, \(a^{2}-a b=b^{2}\). Thus, \(a(a-b)=b^{2}\). We then get \(\dfrac{a}{b}=\dfrac{b}{a-b}\). Both sides are \(\varphi\) .
Fibonacci Sequence
The Fibonacci sequence is a list of numbers. Start with 1, 1, and then you can find the next number in the list by adding the last two numbers together. The resulting (infinite) sequence is called the Fibonacci Sequence. Since we start with 1, 1, the next number is 1+1=2. We now have 1, 1, 2. The next number is 1+2=3. We now have 1, 1, 2, 3. The next number is 2+3=5. The next one is 3+5=8, and so on. Each of these numbers is called a Fibonacci number. Originally, Fibonacci (Leonardo of Pisa, who lived some 800 years ago) came up with this sequence to study rabbit populations! He probably had no idea what would happen when you divide each Fibonacci number by the previous one, as seen below.
|
Fibonacci number |
divided by the one before |
ratio |
|---|---|---|
|
1 |
||
|
1 |
1/1 |
= 1.0000 |
|
2 |
2/1 |
= 2.0000 |
|
3 |
3/2 |
= 1.5000 |
|
5 |
5/3 |
= 1.6667 |
|
8 |
8/5 |
= 1.6000 |
|
13 |
13/8 |
= 1.6250 |
|
21 |
21/13 |
= 1.6154... |
|
34 |
34/21 |
= 1.6190... |
|
55 |
55/34 |
= 1.6177... |
|
89 |
89/55 |
= 1.6182... |
|
... |
... |
... |
|
|
= 1.6180... |
Here is a very surprising fact:
Note
The ratio of two consecutive Fibonacci numbers approaches the Golden Ratio.
It turns out that Fibonacci numbers show up quite often in nature. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone. The Fibonacci numbers are also found in the family tree of honeybees.
Meanwhile, many artists and music researchers have studied artistic works in which the Golden Ratio plays an integral role. These include the works of Michelangelo, Da Vinci, and Mozart. Interested readers can find many resources and videos online. Perhaps it is not surprising that numbers like 3, 5, 8, and 13 are rather important in music theory; just take a quick look at the piano keys!
Reference
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Contributors and Attributions
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Saburo Matsumoto
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