A golden ratio

Muß es sein?

At times I can understand why it made sense to just say that an amazing being beyond our comprehension made the world in seven days; there are some things in the universe that are so curious they are beyond the reaches of human fiction. The quantum world provides an apt example, where uncertainty is an integral feature to the human perception. Indeed it is this notion that has created so many currently irreconcilable issues with the relativistic universe, where uncertainty and space-time sit like oil and water. Today the subject of discussion is a little less ambitious than the quantum universe, but packed with fascination and wonder. Here is the subject for today’s discussion.

\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .

For the rest of this discussion, consider phi to represent the above number; the golden ratio.

Origins

As with many curious things, the thing is discovered before the full curiosity is realised. As this article develops, you will see that discovered is the right word; try not to be too human and stake the existence of something upon your own perception, this feature of the universe is (almost) as old as space and time. Spine tingling.

What is so wonderful about this piece of mathematics is it is stunningly elementary. Take a look at the following very simple diagram.

golden-ratio-diagram

Here I have two lengths; one is length a and the other is length b. The have been divided at a specific point – a specific point such that the following holds:

It does not matter what a and b are, except for the fact they must belong to the following set and obey the above rule. If this does not hold then it is not a golden ratio.

\mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\}

If you think that set notation looks unnecessary that’s because it is; highly unnecessary, I just derive an unhealthy amount of pleasure from sets and over punctuating sentences. Anyhow; I should note a slight caveat; whilst you can convince yourself of the following with any positive real number, to derive the exact surd from you would need to use positive rational number.

Let me underwhelm you first then re-whelm you all over again; this is all the golden ratio is:

Once you do some simple rearranging of this number it is quite easy to derive the following quadratic equation.

And from here it is no great struggle to get the answer above;

\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .

There is of course a solution where we minus the root five and end up with a negative number but this is discarded in context. So this concludes our very underwhelming discussion of where the golden ratio comes from; it is a simple quadratic derived from the ratio of lengths. Now the fat is chewed, let us feast.

Geometry

φ turns up all over the place in geometry; in fact you will have come into contact with this golden ratio many times without realising it. You may have even been attracted to the golden ratio, and just thought they were nice facial features. Here are some golden ratio spirals:

goldenratiospiral

As you can see the succession of squares follows the pattern of beginning with sides of 1/φ being progressively multiplied by 1/φ as illustrated, with a quarter circle within each square, tangent to the edges. Now up to this point the construction may have felt fairly human; however the spiral described above obeying the golden ratio is pervasive through the universe we live in. Take a look at the following image.

20141024174059-spiralsinnature

In that image you are looking at plants, seashells, cloud spirals and galaxies; fundamentally worlds apart yet unified. The central spiral you can see, when overlayed with the golden ratio spirals are found to be in agreement. Remember how to find out if something is real? You play nature knows – and it seems nature knows the golden ratio very well.  Indeed some of the worlds most famous artworks are said to be a slave to little more than the golden ratio of spirals. I think when you find a number that can be used to link a seashell to a galaxy, it deserves a little bit of your respect.

The Fibonacci sequence

I like the Fibonacci sequence a lot; I think it is quite a “natural” sequence; in the sense that it makes sense to add numbers in that way. As it turns out, the Fibonacci sequence and the golden ratio are the very best of friends. You can actually come up with a rough approximation (it always rounds to the correct number) for the Fibonacci sequence using the following formula (sorry I was having a LaTeX issue).

In fact we can do better – I am sure you are relieved to know. Rough approximations may be good enough for a cheap party trick but it does not cut the mustard for the mathematician. We can actually get a little more precise than this with the following very interesting fact was proven by  Kepler as shown below.

\lim _{n\to \infty }{\frac {F(n+1)}{F(n)}}=\varphi .

Now that is just wonderful; as n increases in size, the ratio of the Fibbonaci number over its predecessor is the golden ratio; that is as n tends to infinity the ration tends to gold. There it is again, embedded in the universe in an area you would not expect to find it. If this isn’t getting you excited then I don’t know what will. It turns out there are many different sequences and relationships for which the golden ratio holds true.

Other areas of interconnection

Now that we have had a few examples of what the golden ratio actually is, to save me writing thousands of words I am now going to give you some examples of where the golden ratio has been observed.

  • The diameters of the Earth and the Moon form a triangle based on the phi construction of a triangle.
  • Many of the pyramids of Egypt show the same.
  • It is found that the human face obeys the constructions of the golden spirals; the closer the agreement the more aesthetically pleasing.
  • The pentagon is completely packed with the number.
  • In the spirals of your DNA.

If this was not enough for you, you can write the golden ratio as a nested rational; gorgeous.

 phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...))))

That is all I will write on this number; I am sure the majority of this is not new, but it is never a bad thing to remember a number connecting the stars with your DNA. You did come from the stars after all.

Es muß sein!

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20 responses to “A golden ratio

  1. Pingback: The Fibonacci Series – I can't believe it!·

  2. Now is it a strange coincidence that yesterday I randomly came across this https://www.youtube.com/watch?v=qhOwucw-GRg which introduced me for the first time to the golden ratio and the spectacular structures in South Aftrica that have been largely ignored… and then today I receive your ‘like’ of my thought for Thursday, which led me to this post about the golden ratio… I suspect that’s some sort of ‘coincidental message’ I was meant to receive… Thank you for leading me here… x (The last part of the Youtube clip above does get a bit political but I found the first half fascinating.. )

    Liked by 2 people

    • Thank you very much for the link I shall watch it when I am not stuck at my desk! I am glad you enjoyed the post, it is a very nice area of mathematics which leaks into so many areas of the real world

      Liked by 1 person

      • I have to say – Maths was never a good subject for me at school (many moons ago) so I couldn’t follow the algebra.. however, the post was really interesting – Thank you and I hope you enjoy watching the link.. Enjoy your day x

        Liked by 1 person

  3. Pingback: What is the Golden Ratio? | Novus Lectio·

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