Thanks for submitting this. After seeing the article that claimed the design of the iCloud icon was brilliant because it included Fibonacci ratios, I desperately wanted to nail this article to the (other) author's monitor.
Whenever I see that kind of unjustifiable Fibonacci-worship, I get the same feeling I do when I read creation myths or just-so stories. It honestly hurt the first time I read the truth, and realized I was taught many of the outlandish claims this author points out as fact in grade school math classes.
I agree. Most books and articles on Fibonacci love don't provide enough data to verify the claims. I find the sequence fascinating, but I hold my wallet when someone claims they use it to predict the stock market.
I don't know about all that, but my math background does nudge me to point out one thing:
The fibonacci sequence is such that a[n+2] = a[n+1] + a[n], and as a result, we have
a[n+1] / a[n] -> a[n+2] / a[n+1]
that is to say, the sequence of ratios a[n+1] / a[n] tends to a limit, and this limit is Phi
here is the cool thing: in spirals, there is a certain scale invariance going on when it comes to fibonacci numbers. Consider a rectangle where its larger side is Phi * its smaller side. If whatever process made the rectangle uneven now operates with the larger side, it will make a rectangle whose larger side is Phi * its smaller side (assuming the process is scale invariant). Of course, in reality this would be much more of a continuous process. Put another way, if a process has the following properties:
1) it is scale invariant
2) it is rotation invariant
3) its operation on the larger structure overpowers its operation on the smaller structure
then this process would produce something akin to a golden ratio spiral. So I would expect at least some things in nature to be like this, at least theoretically.
EDIT: that said, passing the golden ratio thing off as fact when it's just a coincidence is doing a big disservice to science, especially when done by science teachers.
This is the mechanism for pinecones, sunflower seeds and tree branches. As far as I'm aware, this mechanism is the only place in nature where Phi, or rather the angular ratio 137.5 : 222.5 degrees, occurs.
If you know of a significantly different mechanism in nature where Phi occurs, I'd love to hear about it.
Please note that the Fibonacci sequence (as well as other less apparent sequences) are a side effect of this angular ratio, if the specimen develops flawlessly. Yet even if the specimen has flaws (due to external factors), it will still grow and tend to return towards the 137.5:222.5 angular ratio, even though it will never return to following the Fibonacci sequence numbers.
It is nice when someone points these things out. Good article.
However, it should be noted that not all appearances of the Fibonacci sequence are the result of misunderstandings & fakery. This is noted in the article:
Phylotaxis. .... This is one part of nature where the fibonacci sequence and related sequences seem to show up uncommonly often, and it's legitimate to inquire why. The interesting cases are seedheads in plants such as sunflowers, and the bract patterns of pinecones and pineapples.
the fibonacci sequence is a side effect of these patterns, not the other way around. if they get disturbed, as they very often do in nature, they no longer follow the fibonacci sequence. the angles between branches on a tree do often tend to the golden ratio (137.5 degrees) and will continue to do so after a disturbance.
fact of the matter is that only the perfect, flawless examples follow a fibonacci sequence.
Heh, I have to wonder who drew that diagram of the Platonic solids corresponding to elements. (Kepler? Looking it up, apparently so.) What's funny is not the mysticism of it, but that the mysticism has been done in a blatantly arbitrary fashion -- if it's to be even slightly credible, obviously the tetrahedron and the dodecahedron should be swapped, so that fire is dual to water and so that the ether is self-dual!
The golden ratio, rule of thirds and other such formulas for creating aesthetically pleasing images probably work because they cause the artist to actually stop & think about composition, kind of like how feng-shui causes you to think harder about the arrangement of your furniture than if you just put it wherever. No mystical forces involved.
As for spirals, well they just look pretty. All swirly & hypnotic...
As an occasional semi-pro photographer and graphic designer, I agree. And this view can be extended to other things like religious and political beliefs. I know plenty of people who hold ethical views I strongly disagree with, but who act in an overall remarkably ethical way because holding their views strongly forces them to keep an eye on ethics at all times.
Meh... what I've always gotten out of stuff about Fibonacci numbers is that it's pretty amazing that a lot of disparate things in nature seem to arrive at similar patterns because very different situations end up having similar sorts of constraints.
The fact that the Fibonacci sequence doesn't explain the mechanism is kinda the beauty of the thing.
Sure people take it too far, or misinterpret it and what not... but that doesn't stop this article from being a little silly.
Except that if you look at what processes in nature really arrive at Phi, they are not that disparate at all.
Which is why it's important to point out that Phi does not occur in the spiral of a nautilus shell or a galaxy or in fact any spiral.
But it does occur in a very specific type of growth mechanism, in the form of an angular ratio of 137.5:222.5 degrees.
And only there.
Which makes Phi just about as equally amazing as the exponential function, squares of integers, triangular numbers, limited growth sigmoid function, the Feigenbaum constant, spheres, circles, symmetry, lines or just about anything else where something in Mathematics explains a tiny part of nature ;-)
Very interesting reading. thank you =)
Thanks for submitting this. After seeing the article that claimed the design of the iCloud icon was brilliant because it included Fibonacci ratios, I desperately wanted to nail this article to the (other) author's monitor.
Whenever I see that kind of unjustifiable Fibonacci-worship, I get the same feeling I do when I read creation myths or just-so stories. It honestly hurt the first time I read the truth, and realized I was taught many of the outlandish claims this author points out as fact in grade school math classes.
They were just fibbing.
I agree. Most books and articles on Fibonacci love don't provide enough data to verify the claims. I find the sequence fascinating, but I hold my wallet when someone claims they use it to predict the stock market.
I don't know about all that, but my math background does nudge me to point out one thing:
The fibonacci sequence is such that a[n+2] = a[n+1] + a[n], and as a result, we have
a[n+1] / a[n] -> a[n+2] / a[n+1]
that is to say, the sequence of ratios a[n+1] / a[n] tends to a limit, and this limit is Phi
here is the cool thing: in spirals, there is a certain scale invariance going on when it comes to fibonacci numbers. Consider a rectangle where its larger side is Phi * its smaller side. If whatever process made the rectangle uneven now operates with the larger side, it will make a rectangle whose larger side is Phi * its smaller side (assuming the process is scale invariant). Of course, in reality this would be much more of a continuous process. Put another way, if a process has the following properties:
1) it is scale invariant 2) it is rotation invariant 3) its operation on the larger structure overpowers its operation on the smaller structure
then this process would produce something akin to a golden ratio spiral. So I would expect at least some things in nature to be like this, at least theoretically.
EDIT: that said, passing the golden ratio thing off as fact when it's just a coincidence is doing a big disservice to science, especially when done by science teachers.
First of, Phi is the limit ratio of many sequences, not just the Fibonacci sequence.
Now I don't quite follow your rectangle example, but this is the process of how and why Phi occurs in nature:
http://www.wolframscience.com/nksonline/page-410
This is the mechanism for pinecones, sunflower seeds and tree branches. As far as I'm aware, this mechanism is the only place in nature where Phi, or rather the angular ratio 137.5 : 222.5 degrees, occurs.
If you know of a significantly different mechanism in nature where Phi occurs, I'd love to hear about it.
Please note that the Fibonacci sequence (as well as other less apparent sequences) are a side effect of this angular ratio, if the specimen develops flawlessly. Yet even if the specimen has flaws (due to external factors), it will still grow and tend to return towards the 137.5:222.5 angular ratio, even though it will never return to following the Fibonacci sequence numbers.
I think this is the type of process I was talking about. And yes, it's Phi that is important, and not the fibonacci sequence.
It is nice when someone points these things out. Good article.
However, it should be noted that not all appearances of the Fibonacci sequence are the result of misunderstandings & fakery. This is noted in the article:
the fibonacci sequence is a side effect of these patterns, not the other way around. if they get disturbed, as they very often do in nature, they no longer follow the fibonacci sequence. the angles between branches on a tree do often tend to the golden ratio (137.5 degrees) and will continue to do so after a disturbance.
fact of the matter is that only the perfect, flawless examples follow a fibonacci sequence.
Heh, I have to wonder who drew that diagram of the Platonic solids corresponding to elements. (Kepler? Looking it up, apparently so.) What's funny is not the mysticism of it, but that the mysticism has been done in a blatantly arbitrary fashion -- if it's to be even slightly credible, obviously the tetrahedron and the dodecahedron should be swapped, so that fire is dual to water and so that the ether is self-dual!
The golden ratio, rule of thirds and other such formulas for creating aesthetically pleasing images probably work because they cause the artist to actually stop & think about composition, kind of like how feng-shui causes you to think harder about the arrangement of your furniture than if you just put it wherever. No mystical forces involved.
As for spirals, well they just look pretty. All swirly & hypnotic...
As an occasional semi-pro photographer and graphic designer, I agree. And this view can be extended to other things like religious and political beliefs. I know plenty of people who hold ethical views I strongly disagree with, but who act in an overall remarkably ethical way because holding their views strongly forces them to keep an eye on ethics at all times.
This article soooo needed writing.
There are people building entire careers out of financial Fibonacci fakery.
Seems like the most common occurrence of the fibonacci series in nature is in bad programming language benchmark code.
Actually, because 1 mile is about 1.609344 km, you can convert approximately between miles and km using Fibonacci numbers. See http://www.catonmat.net/blog/using-fibonacci-numbers-to-conv...
And why use fibonacci numbers, instead of say, multiply by 1.6?
Meh... what I've always gotten out of stuff about Fibonacci numbers is that it's pretty amazing that a lot of disparate things in nature seem to arrive at similar patterns because very different situations end up having similar sorts of constraints.
The fact that the Fibonacci sequence doesn't explain the mechanism is kinda the beauty of the thing.
Sure people take it too far, or misinterpret it and what not... but that doesn't stop this article from being a little silly.
Except that if you look at what processes in nature really arrive at Phi, they are not that disparate at all.
Which is why it's important to point out that Phi does not occur in the spiral of a nautilus shell or a galaxy or in fact any spiral.
But it does occur in a very specific type of growth mechanism, in the form of an angular ratio of 137.5:222.5 degrees.
And only there.
Which makes Phi just about as equally amazing as the exponential function, squares of integers, triangular numbers, limited growth sigmoid function, the Feigenbaum constant, spheres, circles, symmetry, lines or just about anything else where something in Mathematics explains a tiny part of nature ;-)
I think the ratio is still magical. Watch Sal's video on this here: http://www.youtube.com/watch?v=5zosU6XTgSY&feature=playe...
Awwww those 149 karma points could've been mine. I posted this link yesterday in the iCloud icon thread.
Oh well, it's good the get this info out there nonetheless. And it's not like I wrote the article or something, anyway.