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| 1 | +--- |
| 2 | +layout: post |
| 3 | +title: Golden Ratio and Fibonacci Numbers |
| 4 | +--- |
| 5 | + |
| 6 | +Pretty much everybody heard about the golden ratio, and if you are a little |
| 7 | +math-inclined or still remember something from school, you may have heard |
| 8 | +the Fibonacci numbers too. That is, 0, 1, 1, 2, 3, 5, 8, 13, 21, etc, where |
| 9 | +each number in the sequence is the sum of the previous two numbers, |
| 10 | +$$f_n = f_{n-1} + f_{n-2}$$. The golden ratio and the Fibonacci numbers |
| 11 | +are related. As the sequence goes on, the ratio of the adjacent numbers |
| 12 | +approaches the golden ratio, i.e., |
| 13 | +$$ |
| 14 | +\lim_{n\to \infty} (f_n/f_{n+1}) = \phi = 0.61803398875 \dots |
| 15 | +$$ |
| 16 | + |
| 17 | +Many people think that the golden ratio is the most beautiful ratio |
| 18 | +and can be found everywhere in art, architecture, and even in nature. |
| 19 | +But some people argue that most of those perceptions are |
| 20 | +[just myths](http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm). |
| 21 | +People often strech the observations to fit their theories and cherry-pick |
| 22 | +evidence. But, I am not going to talk about this here. |
| 23 | + |
| 24 | +On wikipedia or Google you can find a lot of mathematics on the golden ratio |
| 25 | +and Fibonacci numbers. Here I want to share a simple way I understand |
| 26 | +(and remember) them. |
| 27 | + |
| 28 | +Golden Ratio and Paper Cutting |
| 29 | +------------------------------ |
| 30 | + |
| 31 | +The simplest way (for me, at least) to remember the golden ratio is the |
| 32 | +following. Take a rectangular piece of paper; cut away the largest square |
| 33 | +(the side equal to the short side of the original rectangle); if the |
| 34 | +remaining rectangle is *similar* to the original, then ratio of the |
| 35 | +two sides of the rectangle (short to long) is the golden ratio. |
| 36 | + |
| 37 | + |
| 38 | + |
| 39 | +As shown in the picture above, the equality of the two ratios can be |
| 40 | +expressed in an equation: |
| 41 | + |
| 42 | +$$ |
| 43 | +\frac{1-x}{x} = \frac{x}{1}. |
| 44 | +$$ |
| 45 | + |
| 46 | +Multiply both side by $$x$$ and move things around a little bit, you get |
| 47 | + |
| 48 | +$$ |
| 49 | +x^2 + x -1 =0. |
| 50 | +$$ |
| 51 | + |
| 52 | +It's a simple quadratic equation and can be solved easily, |
| 53 | +$$ x = \frac{1}{2}(-1 \pm \sqrt{5}) $$. Taking the positive solution, |
| 54 | +we get $$ x = \phi= \frac{1}{2}(-1 + \sqrt{5}) = 0.61803398875 \dots $$. |
| 55 | +The other solution is exactly $$-1/\phi = -1-\phi = -1.61803398875 \dots $$. |
| 56 | + |
| 57 | +Fibonacci Numbers and Convergence of Ratios |
| 58 | +------------------------------------------- |
| 59 | + |
| 60 | +The ratio of adjacent numbers in Fibonacci sequence converges to |
| 61 | +the golden ratio. |
| 62 | + |
| 63 | +Let's prove the ratio will converge. For a sequence $$\{a_n\}$$ where |
| 64 | +$$a_n = a_{n-1}+a_{n-2}$$, let's take the difference of the two ratios |
| 65 | + |
| 66 | +$$ |
| 67 | +\begin{eqnarray} |
| 68 | +\delta_n &=& \frac{a_{n+1}}{a_{n}} - \frac{a_{n+2}}{a_{n+1}} \\ |
| 69 | +&=& \frac{a_{n+1}^2 - a_{n}a_{n+2}}{a_n a_{n+1}} |
| 70 | +\end{eqnarray} |
| 71 | +$$ |
| 72 | + |
| 73 | +The numerator can be written as |
| 74 | + |
| 75 | +$$ |
| 76 | +\begin{eqnarray} |
| 77 | +N_n &\equiv& a^2_{n+1} - a_n a_{n+2} \\ |
| 78 | +&=& a^2_{n+1} - a_n (a_n + a_{n+1})\\ |
| 79 | +&=& -a_n^2 + a_{n+1}(a_{n+1} - a_n) \\ |
| 80 | +&=& -a_n^2 + a_{n+1}a_{n-1}, |
| 81 | +\end{eqnarray} |
| 82 | +$$ |
| 83 | + |
| 84 | +which is exactly $$-N_{n-1}$$. That is, the numerator of $$\delta_{n-1}$$ |
| 85 | +with a negative sign. So, we know the numerator of $$\delta_n$$ is a |
| 86 | +constant with a flipping sign. But the denominator $a_na_{n+1}$ is a fast |
| 87 | +growing number. So $$\delta_n$$ approaches zero very quickly. Therefore, |
| 88 | +we can conclude that the ratio $$a_{n+1}/a_n$$ converges. |
| 89 | + |
| 90 | +Note that, it does not matter what the starting values $$a_1$$ and $$a_2$$ are. |
| 91 | +As long as $$a_n = a_{n-1}+a_{n-2}$$, the ratio converges as $$n\to \infty$$. |
| 92 | + |
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