What makes the Fibonacci sequence and its related concepts golden?

[Bogrune]

I've been curious about the Fibonacci sequence for quite a while now, so I decided to study it on my own. I noticed that you can get the sequence by adding the numbers on the pascal triangle diagonally, or by simply adding the number that precedes the next (0,1,1,2,3,5,8...). I then watched a video that gives a quick lecture on it, and it mentioned the golden spiral, the golden ratio and the golden rectangle. What I'm pretty curious about is that why are these called "golden?" Does the sequence really have so many applications?

 

[Cerlid]

I think people get misty eyed because they seem to have a lot of contingents to natural processes, and biology in particular.

I suppose in early science and maths and philosophy, it was like discovering God in the machine of life, hence it is given a great deal of respect. To me it's a little more mundane but then I live now not 400 or x thousand years ago...

It's kinda like discovering the philosophers stone, a piece of reality that denotes how everything in a field may work.

In nature though it really says a natural process will often find the most simple solution which will in turn aid it's survival by being simple. Not so much God as just plain old - well I wouldn't call it lazy but efficiency. In physics, biology and chemistry, energy tends towards the most energy efficient simple constructs unless other systems dictate otherwise. A bee doesn't design hexagonal honey combs because it is a mathematical genius, it does so because they are the easiest to make. Snowflakes aren't complex and unique to amaze people, they are because that is how water works efficiently in its environment.

 

[Shaky]

The concept of a limit can be brought into question using the idea of the golden ratio. Take any real number and sequence it towards Phi. (Add the denominator to the numerator and make that the numerator in the next sequence while the numerator of the last sequence becomes the denominator of the next.)

As the number of sequences tend to infinity, the number in the sequence tends to Phi, however, it can never actually equal Phi, for if it did its value could be expressed as a fraction and therefore not irrational.

 

[lavinia]

I then watched a video that gives a quick lecture on it, and it mentioned the golden spiral, the golden ratio and the golden rectangle. What I'm pretty curious about is that why are these called "golden?" Does the sequence really have so many applications?

The successive ratios of adjacent pairs of number in the Fibonacci sequence converges to the Golden ratio.

You can see these by translating these ratios into the continued fractions

1, 1+1/1+1 ,1+ 1/ 1+1/1+1 ,...

I think - but don't know - that the ancient Greeks thought that the Golden ratio was the most pleasing proportion of a rectangular frame to view a picture of a facade of a building. For instance, I think that the ratio of the height and width of the facade of the Parthenon is the Golden Ratio. It would be interesting to know of the Greeks assigned some higher significance to this. It would not surprise me if they did.

 

[Michael Redei]

It would be interesting to know of the Greeks assigned some higher significance to this. It would not surprise me if they did.

The Pythagoreans used the regular pentagram as one of their mystic symbols, and that is quite full of examples of the golden ratio: http://en.wikipedia.org/wiki/Pentagram#Geometry

 

[Redbelly98]

The concept of a limit can be brought into question using the idea of the golden ratio. Take any real number and sequence it towards Phi. (Add the denominator to the numerator and make that the numerator in the next sequence while the numerator of the last sequence becomes the denominator of the next.)

The successive ratios of adjacent pairs of number in the Fibonacci sequence converges to the Golden ratio.

Sounds like you two are talking about the same thing.

 

[coolul007]

The Greeks assigned the Golden Ratio to all of there statues, the nose divides the face, the belly button divides the body, etc They used the Golden Ratio for these. Plus in any kind of art, the ratio is "pleasing" to the eye. A lot of picture frames use the adjacent Fibonacci numbers as sides, 5 x 8.

 

[Quantum mom]

I suppose in early science and maths and philosophy, it was like discovering God in the machine of life, hence it is given a great deal of respect. To me it's a little more mundane but then I live now not 400 or x thousand years ago...

I'm not as smart as all you are. I'm simple but like asking questions so forgive me my ignorance. Let's for a moment assume God is real. I think HE hides things because HE takes great pleasure out of "man" seeking it out! It's not a bad thing. I think it's a connection to the ultimate unseen realm! It reminds me of the binary code.. 0 off 1 on! Wave function with no observation off or observation wave function collapses= particle on.

 

[Quantum mom]

Like I said. I like to strip it down to "what does this really mean".

 

[Quantum mom]

The concept of a limit can be brought into question using the idea of the golden ratio. Take any real number and sequence it towards Phi. (Add the denominator to the numerator and make that the numerator in the next sequence while the numerator of the last sequence becomes the denominator of the next.)

As the number of sequences tend to infinity, the number in the sequence tends to Phi, however, it can never actually equal Phi, for if it did its value could be expressed as a fraction and therefore not irrational.And theirin lies the question. You can't rationally say that the Fabonacci scale isn't accurate. Phi is a legitimate formula. If you got something better then let's hear it! Otherwise...well you know!

 

[Quantum mom]

The concept of a limit can be brought into question using the idea of the golden ratio. Take any real number and sequence it towards Phi. (Add the denominator to the numerator and make that the numerator in the next sequence while the numerator of the last sequence becomes the denominator of the next.)

As the number of sequences tend to infinity, the number in the sequence tends to Phi, however, it can never actually equal Phi, for if it did its value could be expressed as a fraction and therefore not irrational.





And theirin lies the question. You can't rationally say that the Fabonacci scale isn't accurate. Phi is a legitimate formula. If you got something better then let's hear it! Otherwise...well you know!

How can non local QM wave be materialized as a solid particle w proportions with mathematical identities that go (in the scale of matter) into the micros all the way to the macros cosmos? I'm just asking?

 

[nitsuj]

How can non local QM wave be materialized as a solid particle w proportions with mathematical identities that go (in the scale of matter) into the micros all the way to the macros cosmos? I'm just asking?

Given your previous posts, is that God you're asking?

Can you reword the first sentence. What is a "QM wave"? and how does a [EM?] wave "materialize" & what does "materialize" mean? And then become a solid particle? What is a solid particle? What are "w proportions"? What does it mean to go from micros to macros cosmos? What's micro & what's macro? What's "mathematical identities"?