Toon: "I don't know how often you guys check mail, so this might well look like I need something better to do. And that'd be absolutely correct; I'm bored as hell.

Toon: Anyway! I find the whole discussion you're having on the number 23 very interesting myself. And it's even more interesting that you mentioned the Fibonacci sequence and the golden section. It just so happens that I wrote my "end of high school major coursework" (I don't have a name for it in Dutch, so the discription will have to do) about that very subject.

Toon: I basically spent over 6 months of my life studying those two things, and seeing where and how they occur in nature, art, photography and even music. The site you linked was one of my major sources. It was definitely my number one internet source. I can't remember exactly how much that site explains, but you'll find that the golden section is in fact the limit of the ratio of the subsequent Fibonacci number. This link is important, I'll come back to that in a second.

Toon: E.g.
1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5 / 3 = 1.666...
8 / 5 = 1.6
...
28657 / 17711 = 1.61803399
46368 / 28657 = 1.61803399
...
? / ? = 1.618033988749... = Phi

Toon: Phi (capital P) being the golden ratio, with phi = 1 / Phi = 0.618033988

Toon: Also, Phi = 1 / phi = 1 + phi. That makes phi a curious number in mathematics, much like the relation e ^ (i * π) = -1 is curious. But it has these interesting properties because of its very difinition: P(P - 1) = 1.

Toon: Now, let's look at Phi and phi in nature. This is simply the case because a lot (A LOT) of growth involves some form of spiraling, and the most effective spiral packing is to have the next leaf (or whatever else) be as big as the previous two together. This means the Fibonacci series is involved, and so also Phi. There's nothing mystical or mythical about that, it's simply nature being efficient as a result of millions of years of evolution.

Toon: Bee hives use hexagons for the exact some reason: efficiency.

Toon: The golden section is sometimes used in art and photography. This has to do with where the eyes first focus on a rectangle. This was originally approximated as 2/3rd, but was later said to be phi by some people intrigued by this number, but the only thing you can do with phi in art is create and efficient spiral. Any other percieved uses are arbitrary, and the 2/3rd approximation will do you just as well. Using phi or Phi extensively in art will not lead to a more beautiful image, and will instead in general just look awkward.

Toon: As far music involving phi, there's only one case of a musician using phi (I'd have to go look up which one) and it was completely arbitrary (he was messing with the number in his notes). No other songs, instrument or anything else involves phi.

Toon: I guess the main point of these ramblings is that phi appears in nature because it is efficient, not because it is "special", and trying to give meaning to phi outside of efficiency and mathematical tricks is not a good idea."

Marcus: Phew. Many thanks for that Toon, it makes for interesting reading.

justin: I am aware of the reason why the spiral ratio is used in nature - I'm not trying to suggest it is mystical. Nor am I trying to suggest that 23 is 'mystical'. "Mysticism" is a bit of an insult in these cases - however, as far as Marcus is concerned, you are simply obsessed by numbers and that invalidates anything you have to say. Also, I disagree that the only thing you can do in art with phi is create a spiral. Architects have used it for centuries and it's considered a harmonious form... you could say it's completely arbitrary, but sometimes what we consider beauty is based on succesful ratios and there is a precise formula - eg. here. also, I thought traditional western muscal scales were derives from the ratio (with one or two changes). Here. So maybe it might be worth looking into some more eh Toon?

Marcus: I think Toon perfectly highlights a rational explanation for the occurrence of the numbers 2, 3 and 23 in nature. However, I view any individual's anecdotal observations claiming an increased significance of any number cropping up more than it should ("synchronicity" events etc) with suspicion and a likely sign of number fixation.

toon: To Justin's response which links to http://goldennumber.net/: I would just like to say that that site also claims that Phi can be used to predict stocks (http://goldennumber.net/stocks.htm) and appears in currency exchange (http://goldennumber.net/forex.htm). 'Nuff said.

toon: Like I said, I had to do research for a school project, and I did come across pretty much every site that has some information about Phi or the Fibonacci numbers. The only places where Phi undoubtedly appears is in spiral growth and mathematics. Any other appearances are approximations with hugely varying accuracy, and in art they are mostly arbitrary.

toon: You and I will never agree on this thing :P In my opinion, this is no different than seeing Jesus in a peanut or a burnt piece of toast.

toon: P.S. I don't mean to be insulting, but this kind of thing gets of my nerves much like religion does, and it's sometimes hard to conceal my very vivid feelings about the subject.

justin: Hmm, gets on your nerves eh? sounds like you and the religious both have the same problem when it comes to objectivity. But Toon, just because that website is featuring stocks and currency exchange doesn't mean everything else it says is wrong. It sounds like they are collecting together information - but then, the same can be said of the Monte-Carlo equation, which I use for global illumination rendering, but which can be used for all sorts. Including, I believe, crowd movement and finance. so don't be so quick to dismiss it. You saying phi's use in art as arbitrary merely opens up a whole other can of worms.

toon: Looks like we aren't done with this yet ;P First of all, me and the religious don't "have the same problem when it comes to objectivity". There is no objective proof for the importance of Phi outside the two things I mentioned. Let alone the number 23. The religious tend to believe in something unprovable, but me, I happen to *not* believe in something for which there is no solid evidence.

toon: Monte Carlo is "a class of computational algorithms for simulating the behavior of various physical and mathematical systems". 23 and Phi are both numbers. I fail to see the similarity.

toon: I think it would be best to agree to strongly disagree on this now, before we start comparing people to worse things than the religious. Though, speaking about religion, this does feel a bit like discussing the existence of some sort of god.

justin: Agreed. Altough, the reason why I mention ed Monte-Carlo is because as a "system" it is also used to predict the flow of money... and you used that argument as a reason to discredit Phi.