The Golden Cut!

Today, I read an (old, but) interesting book on “Fibonacci Numbers” (only about 70 pages). It was mathematically intense most of the time, proving theorems and other boring stuff. But, the results were nice & fun to read. A few of my old questions were answered today, and it’s surprising that they were connected by Fibonacci numbers.

This is my favorite video, it’s the one I downloaded (for the first time) from YouTube to watch it again.

Anyways, there’s something called the “Golden ratio”. There are many ways to obtain that thing. But, there’s one experimental way. Start generating the Fibonacci numbers using two variables and try dividing the latter by the former, like this…

\lim\limits_{n \to \infty} \frac{f_{n+1}}{f_n} =\varphi

You may find a ratio that starts at 2 for f_1 and f_2 and begins to converge towards 1.618 (…), which is our beloved Golden ratio. I didn’t figure it out until today (partially because it didn’t bother me much).

Firstly, you can generate Fibonacci numbers using this ratio. Simply start at 1, iterate throughout by multiplying 1.618 along the way and round it off to whole numbers. That’s easy compared to the usual way of using three variables (or other algorithms like Binet’s formula) to generate the numbers. Moreover, we have an advantage. The ratio gets accurate as n \to \infty. So, there’s no way that the error can increase for larger numbers.

Then, there’s an aesthetic property – called “Golden Section Rectangles”. That’s what you saw in the video above. It starts by using those section rectangles. Over the centuries, it has encouraged philosophers & mathematicians to describe nature using pure math. But, they’ve failed obviously. Science is simple, but not that simple! So, this rectangle can be generated by Fibonacci numbers, or it can be constructed to fill infinite squares. Either way, it’s based on “the Fibonacci problem”.

The idea is to construct a nice rectangle such that it contains squares. In this rectangle, all the squares other than the two smallest ones are different. If they were really irrational, then we can’t talk about the last two. But, the practical cases do have the last two squares…

There’s another property, which is also a nice way to get that value…

The ratio is based on this proportionality of line segments shown to the side, which clearly rearranges to…

\varphi^2-\varphi-1=0

Now, this equation has a positive root which corresponds to the golden ratio,

\varphi= \frac{1+\sqrt{5}}{2}

And, because it’s defined in terms of itself, it’s also the only continued fraction that looks beautiful…

\varphi=1+\frac{1}\varphi \implies \varphi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}}

Even \pi and e are not very interesting when it comes to continued fractions. Anyways, apart from all those, this ratio has a lot of physical influences, both in human minds and in nature. That golden rectangle is often praised as the most aesthetically pleasing one. Because, it has one more property, that the rectangle maintains its proportionality on expansion & contraction.

I don’t know whether you’re aware of this. But, many old architectures, books, even the designs of this 21st century like TV & monitor screens (which explains why the size is measured by its diagonal length), different kinds of cards, suitcases, match boxes, and most of the stuff we use everyday, all these things make use of the golden ratio.

And, that wonderful video shows how some forms of nature have already solved the “Fibonacci problem” and exhibit the golden ratio by their physical appearances. All these days, I had no idea that Fibonacci numbers have such magnanimous interconnections – binomial coefficients, prime numbers, continued fractions, geometry, … It’s simply amazing!

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