Exploring the Fibonacci Sequence

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In summary, the Fibonacci sequence is a numerical pattern that can be obtained by adding the previous two numbers in the sequence. It has many applications, including the golden spiral, golden ratio, and golden rectangle, which are all based on the successive ratios of adjacent pairs of numbers in the Fibonacci sequence. This ratio was considered significant by the ancient Greeks and has been used in many forms of art and architecture. The concept of a limit can also be explored using the Fibonacci sequence and its convergence to the golden ratio. Overall, the Fibonacci sequence is a fascinating and versatile concept that has been studied and admired for centuries.
  • #1
Bogrune
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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?
 
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  • #2
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...

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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.
 
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  • #3
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.
 
  • #4
Bogrune said:
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.
 
  • #5
lavinia said:
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
 
  • #6
Shaky said:
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.)
lavinia said:
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. :smile:
 
  • #7
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.
 
  • #8
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.
 
  • #9
Like I said. I like to strip it down to "what does this really mean".
 
  • #10
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!
 
  • #11
Quantum mom said:
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?
 
  • #12
Quantum mom said:
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? :tongue2:

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"?
 

1. What is the Fibonacci Sequence?

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The sequence is named after Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced it to the Western world in his book Liber Abaci in 1202.

2. What is the significance of the Fibonacci Sequence?

The Fibonacci Sequence has numerous applications in mathematics, science, and nature. It has been used to model the growth of populations, the branching of trees, and the arrangement of leaves on a stem. It also appears in the study of fractals, financial markets, and music.

3. How is the Fibonacci Sequence related to the Golden Ratio?

The ratio of two consecutive numbers in the Fibonacci Sequence approaches the Golden Ratio (approximately 1.618) as the sequence progresses. This ratio has been considered aesthetically pleasing and has been used in art and architecture for its perceived harmonious proportions.

4. What is the formula for calculating the nth term in the Fibonacci Sequence?

The formula is Fn = Fn-1 + Fn-2, where F0 = 0 and F1 = 1. This means that each term in the sequence is the sum of the two previous terms. For example, the 6th term in the sequence, also known as F6, would be calculated as F6 = F5 + F4 = (3+5) = 8.

5. Are there any real-world applications of the Fibonacci Sequence?

Yes, the Fibonacci Sequence has been used in various fields such as computer algorithms, stock market analysis, and cryptography. It also has applications in biology, particularly in the study of plant growth and phyllotaxis, which is the arrangement of leaves on a stem.

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