Warning: Starting with this blog you may need some additional magic: in order to typeset mathematical formulas I used a script file named TeX THE WORLD. Consequently you may have to enable javascript in your browser (Greasemonkey and the same user script). Sorry about the inconvenience, but this way maths look better.
As it's name states, the golden ratio has everything to do with geometry and aesthetics. It was defined by the following property: "Given a line segment of length c, how can we divide it in two sub-segments of lengths a and b (c=a+b, a > b) such that the ratio of the large over the small equals the ratio of the segment over the large?" Otherwise said:
[; \varphi = \frac{a}{b} = \frac{c}{a} ;]
Now, replacing c by the sum of a and b, we get:
[; \varphi = \frac{1 + \varphi}{\varphi} ;]
hence: [; \varphi = \frac{1+\sqrt{5}}{2} \approx 1.618 ;]
This proportion was deeply engraved in ancient (the Parthenon), Renaissance (Mona Lisa) and modern (The Sacrament of the Last Supper) masterpieces of painters and architects, and all for a reason: mathematics ARE aesthetically pleasing.
And coincidences don't stop here: remember the Fibonacci sequence? Let's consider consecutive terms ratios and place them like this:
And coincidences don't stop here: remember the Fibonacci sequence? Let's consider consecutive terms ratios and place them like this:
[;\frac{1}{1}, \frac{3}{2}, \frac{5}{8}, ... ? ... , \frac{13}{8},\frac{5}{3},\frac{2}{1};]
We get two sequences, an increasing one and a decreasing one, both leading to ... the golden ratio. So, if you want to use golden ratio-like proportions, the Fibonacci numbers are your best choice, no need for tedious square roots.
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