1. With examples of the Fibonacci sequence popping up all over nature, is this progression just as important and common in cosmology?

2.

3. Okay... I am going to have a Duh moment... What is the Fibonacci sequence again and how is it popping up in nature?

4. The Fibonacci sequence is 0 1 1 2 3 5 8 13 21 34 55 89......

It just basically the last number added to make the next number. It's found in plants, animals, biology even in fractals. There seems something almost magical about the way it just keeps popping up.

5. This is a surprisingly good and enjoyable discussion on the fibonacci sequence.

6. I know that the Fib. sequence is deeply related to the Golden ratio and its negative reciprocal. I'm not sure why, but maybe it's because both the Fib. numbers and have to do with the ratio of two figures and their sums (???).

From this, we can fairly say that it's quite a simple concept to derive them from: a+b=c , b+c=d , c+d=e... and so on. Thus, any natural process involving taking a previous outcome as an addend for the next outcome is bound to show Fib. properties. So, it's not necessarily just *the* Fibonacci sequence (0, 1, 1, 2, 3, 5...) but possibly any Fib. sequence that can pop up in nature, as long as it follows the general rule with its own unique seed values. (also, any such sequence regardless of its seed values should relate to the Golden ratio; correct me if wrong).

7. Originally Posted by halorealm
I know that the Fib. sequence is deeply related to the Golden ratio and its negative reciprocal. I'm not sure why, but maybe it's because both the Fib. numbers and have to do with the ratio of two figures and their sums (???).

From this, we can fairly say that it's quite a simple concept to derive them from: a+b=c , b+c=d , c+d=e... and so on. Thus, any natural process involving taking a previous outcome as an addend for the next outcome is bound to show Fib. properties. So, it's not necessarily just *the* Fibonacci sequence (0, 1, 1, 2, 3, 5...) but possibly any Fib. sequence that can pop up in nature, as long as it follows the general rule with its own unique seed values. (also, any such sequence regardless of its seed values should relate to the Golden ratio; correct me if wrong).
Are the Golden Ratio, now that's an interesting one in it's self. Just quite how and why architects discovered that is a most interesting mystery. We have records from ancient Greek architecture being used, you'll even find evidence of this in the Parthenon. But it wasn't until the renaissance until it really became enshrined in architecture.
There was a very famous italien architect, his name escapes me, who wrote some seminal books on classical architecture styles and all about the importance of the Golden Ratio and it's relevance to what is actually beautiful to the eye.

8. Just looked it up, his name was Andrea Palladio.

9. Yes, the Golden ratio is a peculiar constant all on its own. Because of its interesting properties, not only is it in the league of and , but it's also pretty well known outside the realm of mathematics, e.g. art and architecture. In fact, I think most high school art teachers mention it at least once in any ordinary course. Euclid was the first to define it in records we know, calling it the "extreme and mean ratio". Refer to Book 6, Definition 3 in his work Elements. But, as you said, it was probably known before he recorded this, e.g. the Parthenon.

This is the fundamental definition of . It and its negative reciprocal are the only numbers that satisfy . From this equation you can derive a quadratic equation and then use the quadratic formula, giving . Of course, in ordinary geometry, negative ratios don't work (or do they? I don't know), leaving 1.618... the one to take the prize.

And, according to Wikipedia, Fibonacci himself proved that dividing any consecutive Fibonacci numbers would asymptotically approach as . So, why the relationship between and Fib. sequences? Well I think it's because if you go back to that first definition, "the whole is to the greater as the greater is to the less", you'll see that's sort of what the Fib. sequence is doing.

10. Originally Posted by Chrisgorlitz
Just looked it up, his name was Andrea Palladio.
Hmm, the "most influential individual in the history of Western architecture". I couldn't find any credible sources about his work and , but apparently he utilized it in his Villa Rotunda. It's considered the most beautiful and appealing element in art, and that's hard for any designer or artist to miss out on.

11. Originally Posted by halorealm
Originally Posted by Chrisgorlitz
Just looked it up, his name was Andrea Palladio.
Hmm, the "most influential individual in the history of Western architecture". I couldn't find any credible sources about his work and , but apparently he utilized it in his Villa Rotunda. It's considered the most beautiful and appealing element in art, and that's hard for any designer or artist to miss out on.
Yes slightly embassing, I couldn't for the life of me recall Bobby Moore's name the other day while watching 'Escape to Victory', which is almost a shooting offence.
Andrea Palladio wrote 4 books, the I quattro libri dell'architettura, these were very important and they really helped to define the relationstip between aesthetic beauty and the Golden Ratio.

12. It would not surprise me one bit if the Fibonacci sequence is just as common and relevant on a much grander scale as it appears to be here on Earth. We have already been able to observe it to a certain extent in distant spiral galaxies. If we ever succeed in becoming a a more advanced, space-faring civilization, it will be interesting to see where else the Fibonacci sequence is found, and with what frequency. I would hope that such information might put it into a more universal context for us and help us to understand it better.

13. Originally Posted by TheObserver
This is a surprisingly good and enjoyable discussion on the fibonacci sequence.
Cool, doodling in maths.

14. Originally Posted by Saturn
It would not surprise me one bit if the Fibonacci sequence is just as common and relevant on a much grander scale as it appears to be here on Earth. We have already been able to observe it to a certain extent in distant spiral galaxies. If we ever succeed in becoming a a more advanced, space-faring civilization, it will be interesting to see where else the Fibonacci sequence is found, and with what frequency. I would hope that such information might put it into a more universal context for us and help us to understand it better.
It would certainly be interesting to find out, I have this feeling that there is some real, but hidden, significance about the fibonacci sequence that we are yet to understand.

15. Originally Posted by Chrisgorlitz
Originally Posted by Saturn
It would not surprise me one bit if the Fibonacci sequence is just as common and relevant on a much grander scale as it appears to be here on Earth. We have already been able to observe it to a certain extent in distant spiral galaxies. If we ever succeed in becoming a a more advanced, space-faring civilization, it will be interesting to see where else the Fibonacci sequence is found, and with what frequency. I would hope that such information might put it into a more universal context for us and help us to understand it better.
It would certainly be interesting to find out, I have this feeling that there is some real, but hidden, significance about the fibonacci sequence that we are yet to understand.
I have often suspected that myself, but I suppose there is really no way to know for certain until we have a more complete picture of the universe.

16. Originally Posted by Saturn
It would not surprise me one bit if the Fibonacci sequence is just as common and relevant on a much grander scale as it appears to be here on Earth. We have already been able to observe it to a certain extent in distant spiral galaxies. If we ever succeed in becoming a a more advanced, space-faring civilization, it will be interesting to see where else the Fibonacci sequence is found, and with what frequency. I would hope that such information might put it into a more universal context for us and help us to understand it better.
Like cosmological? I suspect everything in math holds some part in the big picture of the universe's "mechanism". It is the ultimate model, the most precise description of everything... or not. But like you said, we need a better understanding before reaching that conclusion.

17. Originally Posted by halorealm
Originally Posted by Saturn
It would not surprise me one bit if the Fibonacci sequence is just as common and relevant on a much grander scale as it appears to be here on Earth. We have already been able to observe it to a certain extent in distant spiral galaxies. If we ever succeed in becoming a a more advanced, space-faring civilization, it will be interesting to see where else the Fibonacci sequence is found, and with what frequency. I would hope that such information might put it into a more universal context for us and help us to understand it better.
Like cosmological? I suspect everything in math holds some part in the big picture of the universe's "mechanism". It is the ultimate model, the most precise description of everything... or not. But like you said, we need a better understanding before reaching that conclusion.
Exactly, cosmological. I share your suspicions about mathematics, as it does seem to be strongly associated with the laws of nature. Have you ever read Contact by Carl Sagan? There is an idea presented in that book which states that there is hidden information about the universe contained within the digits of pi. This discussion kind of reminded me of that.

18. Exactly, cosmological. I share your suspicions about mathematics, as it does seem to be strongly associated with the laws of nature. Have you ever read Contact by Carl Sagan? There is an idea presented in that book which states that there is hidden information about the universe contained within the digits of pi. This discussion kind of reminded me of that.
Sorry, no. I haven't read the book, but I've seen the film. If the movie does have that part, I don't particularly remember it though. However, I have read a novel called Omega Sol, and it's out-of-plot message is that everything is determined by a simple concept: existence and nonexistence, creation and destruction, etc. (like 1's and 0's in a binary code). I found the idea slightly interesting, but I can't really put my heart to it.

As for the mysterious Pi, it's not really what I had in mind. Nonetheless, a cool idea. Have you ever heard the saying or something similar to "The works of Shakespeare are written in Pi"? Note that irrational numbers are non-repeating, non-terminating in their decimal form. If you think about it, shouldn't you be able to find any string of numbers somewhere in that infinite stretch of seemingly random digits? It's not repeating, but it still goes on forever. Any string of information is bound to show up somewhere, thus, the laws of the universe are mathematically embedded in what are known as normal numbers. The author could have been specifically referring to this.

This might be a good video to spark some curiosity: Stephen Wolfram on TED: Computing A Theory Of Everything

19. Originally Posted by halorealm
Exactly, cosmological. I share your suspicions about mathematics, as it does seem to be strongly associated with the laws of nature. Have you ever read Contact by Carl Sagan? There is an idea presented in that book which states that there is hidden information about the universe contained within the digits of pi. This discussion kind of reminded me of that.
Sorry, no. I haven't read the book, but I've seen the film. If the movie does have that part, I don't particularly remember it though. However, I have read a novel called Omega Sol, and it's out-of-plot message is that everything is determined by a simple concept: existence and nonexistence, creation and destruction, etc. (like 1's and 0's in a binary code). I found the idea slightly interesting, but I can't really put my heart to it.

As for the mysterious Pi, it's not really what I had in mind. Nonetheless, a cool idea. Have you ever heard the saying or something similar to "The works of Shakespeare are written in Pi"? Note that irrational numbers are non-repeating, non-terminating in their decimal form. If you think about it, shouldn't you be able to find any string of numbers somewhere in that infinite stretch of seemingly random digits? It's not repeating, but it still goes on forever. Any string of information is bound to show up somewhere, thus, the laws of the universe are mathematically embedded in what are known as normal numbers. The author could have been specifically referring to this.

This might be a good video to spark some curiosity: Stephen Wolfram on TED: Computing A Theory Of Everything
Unfortunately, many of the best parts of Contact were excluded from the movie, including the bit about pi. I have indeed heard that saying and it's true. The part about pi in Contact may very well have been a reference to that notion (it's been a number of years since I've read it, so the details about that part of the book are a little vague). It hadn't occurred to me before. Anyway, the book is definitely worth reading (nearly everything by Carl Sagan is, in my opinion) if that sort of thing interests you. It also contains a number of other intriguing ideas that were left out of the movie.

That was a fascinating TED talk, thank you for linking. I've used both Mathematica and Wolfram Alpha for a number of years, and I remember being thoroughly impressed when I discovered that Wolfram Alpha could compute orbital mechanics problems. Was that last part of the talk about finding a model universe that behaved as ours did mathematically more along the lines of what you had in mind, or have I missed the point entirely?

20. Originally Posted by Saturn
Unfortunately, many of the best parts of Contact were excluded from the movie, including the bit about pi. I have indeed heard that saying and it's true. The part about pi in Contact may very well have been a reference to that notion (it's been a number of years since I've read it, so the details about that part of the book are a little vague). It hadn't occurred to me before. Anyway, the book is definitely worth reading (nearly everything by Carl Sagan is, in my opinion) if that sort of thing interests you. It also contains a number of other intriguing ideas that were left out of the movie.
That is unfortunate. Well, I'll look into it. Thanks.

That was a fascinating TED talk, thank you for linking. I've used both Mathematica and Wolfram Alpha for a number of years, and I remember being thoroughly impressed when I discovered that Wolfram Alpha could compute orbital mechanics problems. Was that last part of the talk about finding a model universe that behaved as ours did mathematically more along the lines of what you had in mind, or have I missed the point entirely?
No, you got it: creating a model universe purely by math. I haven't the slightest idea as to how Stephen Wolfram approaches such a goal but, according to him, he's made some surprising progress. Still, I'm somewhat skeptical. I'll try looking for info on this peculiar work of his.

21. Originally Posted by halorealm
No, you got it: creating a model universe purely by math. I haven't the slightest idea as to how Stephen Wolfram approaches such a goal but, according to him, he's made some surprising progress. Still, I'm somewhat skeptical. I'll try looking for info on this peculiar work of his.
I'm skeptical as well and frankly, still a bit confused as to the precise nature of his research. There is a blog published by Wolfram which may be of help (perhaps starting with this article) that may be useful, though I haven't yet had a chance to thoroughly investigate it. Stephen Wolfram also published a book titled "A New Kind of Science", which probably expands upon some of the ideas he presented in the lecture. If you happen to come across any additional info, do you mind posting it? I would be interested to see it. Thanks.

22. Originally Posted by Saturn
I'm skeptical as well and frankly, still a bit confused as to the precise nature of his research. There is a blog published by Wolfram which may be of help (perhaps starting with this article) that may be useful, though I haven't yet had a chance to thoroughly investigate it. Stephen Wolfram also published a book titled "A New Kind of Science", which probably expands upon some of the ideas he presented in the lecture. If you happen to come across any additional info, do you mind posting it? I would be interested to see it. Thanks.
Thanks for the article. Upon casually reading through, I found that there's a readable online version of A New Kind of Science ( here ) on the Wolfram Science website, the important part being Chapter 9. I understand his view a little better now. He states that the fundamental laws of physics may not be truly fundamental, i.e. axiomatic (like mathematics), but could be produced computationally using technology that could run all possible rules to create virtual networks (some of which he considers candidates).

I'm still clueless as to how the nature of these networks may correspond to physical laws, but then again my scope of knowledge is quite limited. Overall, it's not considered mathematical in nature, but more abstractly rule-based/computational. Still skeptical about this work, but nonetheless it has gotten pretty far with both positive and negative reception from the scientific community.

23. By the way, I'm sorry I've veered the thread off its original purpose. I still have a question about the Fibonacci sequence.

So the explicit formula for finding the th term is .

I've noticed it works for the sequence with seed values 0 and 1. I heard you can derive a formula for any seed values. How would I do that?

24. Originally Posted by halorealm
How would I do that?
I did it as a first year linear algebra problem, it was quite taxing I won't lie. If you would like I can dig through my old assignments and let you know where to start?

25. Originally Posted by halorealm
Thanks for the article. Upon casually reading through, I found that there's a readable online version of A New Kind of Science ( here ) on the Wolfram Science website, the important part being Chapter 9. I understand his view a little better now. He states that the fundamental laws of physics may not be truly fundamental, i.e. axiomatic (like mathematics), but could be produced computationally using technology that could run all possible rules to create virtual networks (some of which he considers candidates).

I'm still clueless as to how the nature of these networks may correspond to physical laws, but then again my scope of knowledge is quite limited. Overall, it's not considered mathematical in nature, but more abstractly rule-based/computational. Still skeptical about this work, but nonetheless it has gotten pretty far with both positive and negative reception from the scientific community.
Thanks for the link. I have also started to read the online version of the book, and while I now feel like I have a better grasp on the nature of his work, I find myself feeling more and more skeptical as I continue reading. Perhaps it's just that I'm not accustomed to his style of writing (especially not in the context of science), but some of what he's saying seems to be bordering on self contradictory. It's probably too early for me to be making assessments like that though, without having read more. Seeing as I'm more familiar with tangible science and computational modeling, I'm finding this book rather difficult to read, let alone, wrap my head around some of these abstract concepts and logic that he is presenting. Did you understand things right away, or did you have to keep reading to make sense and more properly define some of what he's saying?

To address your other question, are you actually looking to derive it yourself, or just understand the process? As TheObserver already stated, deriving it yourself using linear algebra techniques is pretty intense, so you might consider watching this instead: The Fibonacci Numbers Using Linear Algebra.

26. Originally Posted by TheObserver
I did it as a first year linear algebra problem, it was quite taxing I won't lie. If you would like I can dig through my old assignments and let you know where to start?
No need, but thanks anyway Plus, my math knowledge is embarrassingly limited (yes, embarrassingly). A painstaking linear algebra problem isn't what I expected though. I thought it would be something more along the lines of elementary algebra... a simple derivation of a general rule, should I say.

Originally Posted by Saturn
To address your other question, are you actually looking to derive it yourself, or just understand the process? As TheObserver already stated, deriving it yourself using linear algebra techniques is pretty intense, so you might consider watching this instead: The Fibonacci Numbers Using Linear Algebra.
Well, as I said, an intense problem wouldn't be best for me, since I most likely won't understand it. However, I found the video quite helpful. I understand a little better. Thanks.

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