add golden ratio and fibonacci number

Author: gitcheng

.gitignore

@@ -1,4 +1,6 @@
+*~
 _site
+_junk
 .DS_Store
 .jekyll
 .bundle

_config.yml

@@ -3,10 +3,10 @@
 #
 
 # Name of your site (displayed in the header)
-name: Chih-hsiang Cheng
+name: Random thoughts 隨便寫寫
 
 # Short bio or description (displayed in the header)
-description: Scientist
+description: by a scientist with a wandering mind
 
 # URL of your avatar or profile pic (you could use your GitHub profile pic)
 avatar: https://avatars3.githubusercontent.com/u/1696423?v=2&s=460
@@ -52,7 +52,7 @@ baseurl: ""
 
 markdown: kramdown
 highlighter: pygments
-permalink: /:title/
+permalink: /blog/:year/:month/:day/:title/
 
 # The release of Jekyll Now that you're using
 version: v1.0.0

_posts/2014-3-3-Hello-World.md

@@ -1,6 +1,8 @@
 ---
 layout: post
 title: Golden Ratio and Fibonacci Numbers
+category: math
+tags: golden_ratio, Fibonacci, series
 ---
 
 Pretty much everybody heard about the golden ratio, and if you are a little
@@ -34,7 +36,7 @@ following. Take a rectangular piece of paper; cut away the largest square
 remaining rectangle is *similar* to the original, then ratio of the 
 two sides of the rectangle (short to long) is the golden ratio.
 
-![Rectangle division](../images/2014/golden-ratio-1.png)
+![Rectangle division]({{ site.url }}/images/2015/golden-ratio-1.png)
 
 As shown in the picture above, the equality of the two ratios can be
 expressed in an equation:
@@ -65,8 +67,8 @@ $$a_n = a_{n-1}+a_{n-2}$$, let's take the difference of the two ratios
 
 $$
 \begin{eqnarray}
-\delta_n &=& \frac{a_{n+1}}{a_{n}} - \frac{a_{n+2}}{a_{n+1}} \\
-&=& \frac{a_{n+1}^2 - a_{n}a_{n+2}}{a_n a_{n+1}}
+\delta_n &=& \frac{a_{n+1}}{a_{n+2}} - \frac{a_{n}}{a_{n+1}} \\
+&=& \frac{a_{n+1}^2 - a_{n}a_{n+2}}{a_{n+1} a_{n+2}}
 \end{eqnarray}
 $$
 
@@ -81,12 +83,46 @@ N_n &\equiv& a^2_{n+1} - a_n a_{n+2} \\
 \end{eqnarray}
 $$
 
-which is exactly $$-N_{n-1}$$. That is, the numerator of $$\delta_{n-1}$$
-with a negative sign. So, we know the numerator of $$\delta_n$$ is a 
-constant with a flipping sign. But the denominator $a_na_{n+1}$ is a fast
-growing number. So $$\delta_n$$ approaches zero very quickly. Therefore,
-we can conclude that the ratio $$a_{n+1}/a_n$$ converges.
+which is exactly $$-N_{n-1}$$, that is, the numerator of $$\delta_{n-1}$$
+with a negative sign. You can go all the way down to $$\delta_0$$; each 
+time you go one step down, you flip the sign but keep the magnitude constant.
+The denominator $$a_{n+1}a_{n+2}$$, on the other hand, is a fast
+growing number. So $$\delta_n$$ will approach zero very quickly. Therefore,
+we can conclude that the ratio $$a_n/a_{n+1}$$ converges (as long as 
+$$\delta_n$$ approaches zero faster than $$1/n$$, which is true in this case).
 
 Note that, it does not matter what the starting values $$a_1$$ and $$a_2$$ are.
-As long as $$a_n = a_{n-1}+a_{n-2}$$, the ratio converges as $$n\to \infty$$.
+As long as $$a_n = a_{n-1}+a_{n-2}$$ relation holds, the ratio converges as 
+$$n\to \infty$$. So it can be regular Fibonacci numbers 
+$$\{1, 1, 2, 3, 5\dots\}$$, or so-called Lucas numbers
+$$\{2, 1, 3, 4, 7\dots\}$$, or whatever $$a_1$$, $$a_2$$ you choose,
+even negative numbers. The ratio converges very quickly.
+
+![Sequence ratio convergence]({{ site.url }}/images/2015/fibonacci-ratio-converge.png)
+
+
+Paper cutting again
+---------------------------
+
+In the first figure, we cut a square out of a rectangular paper, and
+we let the remaining smaller rectangle be similar to the original, and
+then we conclude the ratio of the two sides is the golden ratio.
+Clearly if we start with the golden ratio, we can repeat this operation
+and cut the paper (mathematically) forever.
+
+What if the ratio is not the golden ratio to start with?
+
+First we define the situation that we say we cannot cut the paper in
+way anymore. That is, when the ratio of the two side $$x \ge 2$$, because
+when this happens, the long side will still be a long side after a
+square is removed, or it will become a square and you don't know how to
+cut it anymore.
+
+It can be shown that to cut this rectangular forever, the only ratio
+you can start with is the golden ratio. If you start with a ratio of two
+adjacent Fibonacci numbers (can be very close to the godel ratio but not
+exactly), you will end up with a square, that is, the ratio of the sides
+is $$a_2/a_1$$.
+
+