Golden ratio?

Do any maths enthusiasts know of the Golden ratio? It is a number phi defined by Euclid as phi=1.6180339887. It appears in the natural world in sunflower florets, crystals to the shape of galaxies. Nature uses this number, like pi, to build these shapes. We use this number in architecture - the pantheon, art - the Mona Lisa and music.

What I don't understand is why does it take this particular value?

Reply 1

Phichi

Original post by Son of Zeus

Because it satisfies the equation that the golden ratio implies.

\frac{a+b}{a} = \frac{a}{b} = Phi

By simple manipulation of the formula

\frac{a+b}{a} = \frac{a}{b}

You can form a quadratic in Phi, and solve to find the two values of phi. One of which is negative so is ignored. I'm sure you can google the method to do this if you're further interested. Its like asking, why does Pi take the its particular value, same concept.

(edited 10 years ago)

Reply 2

Jkn

Original post by Son of Zeus

We can derive it directly from its ratio definition (check out here for things like that) which, for me, makes it seem less arbitrary. I think the reason it arises so much in nature as well as many seemingly unrelated pieces of mathematics is because the simple structure that underpins the Fibonacci sequence (which might as well be married to the golden ratio) is based in recursive replication which mimics what goes on in cells.

We see it in plants and life because natural evolution produces mathematically simple solutions to problems as these are the most likely to arise through genetic mutation. Also, more specifically, it is desirable for intelligent animals to have some sort of way of telling whether or not a person's cells are replicating correctly (as they would be more likely to be healthy). As the golden ratio is mathematically simple (and is a structurally 'low-energy' state) this ratio is what we evolved to find desirable. We humans interpret this as a form of visual beauty - which, in today's society, has uses that extend far beyond telling whether or not somebody in healthy!

The Mona Lisa was painted using it as D'Vinci was the first to discover this link between the number and our perception of beauty and things like crystals would have the number by virtue of the simple structure it creates. Another interesting similar ratio, when dealing with structure, is the hexagonal lattice. This ratio helps to answer the question "what is the shortest network of lines I need to link the 4 vertices of a square?" - many are surprised that this solution is non-trivial. This, incidentally, explains why squashed soap bubbles form hexagons, or why bee's honey forms in hexagonal shapes, or why the giants causeway is the way it is!

In terms of its constant appearance in mathematics, I believe this is solely down to its simplistic structure. If you do not find the 'folding rectangle' definition to be particularly un-arbitrary and elegant, why not consider its continued fraction representation which comes directly by virtue of it satisfying the quadratic

x=1+\frac{1}{x}

. In addition to this, it has some rather satisfying algebraic properties (for example, if you square it).

Whilst this explains those things (though I believe it would require things like fractal geometry to justify rigorously - so don't worry if you're not particularly convinced), I like to think that there is still some mystery to the number. Many misrepresent it by saying that it has something to do with religion or 'divine creation' (or whatever?) but, whilst I personally do not see the need for such an appeal, I am still startled when I will be working on a seemingly unrelated maths problem and then suddenly.. and entirely unexpected.. there is is! The golden ratio:

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

Hope this helps!

Reply 3

Son of Zeus

Original post by Jkn

x=1+\frac{1}{x}

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

Hope this helps!

That's very informative. Thank you.

With all what you've said I still don't understand why a Pentagon is the shape nature uses as the lowest energy shape. I can't help but wonder what deeper meaning this number has. What does that tell us about the very fabric of reality? Had the most efficient shape been a square then would our universe look and function very differently. This number seems to be a part of a much larger framework of our universe.

There is a poem I found called the Fibonacci Quarterly. It describes the algebraic properties the number phi has. It reads as follows:

The golden mean is quite absurd;
It's not your ordinary surd.
If you invert it (this is fun!).
You'll get itself reduced by one;
But if increased by unity,
This yields its square, take it from me.

I find these properties astonishing. Phi seems to have a lot more going for it than pi but this may be due to my lack of understanding.

Could you recommend any further reading material on this number. In particular its significance in nature and why it is the most energy efficient number.