I agree with Hal. It's an interesting number, but there's far too much hype about it. It has some geometric significance, so some of the things it applies to there will obviously carry over elsewhere.
"Defined irrational numbers" are not rare. Ones that have enough history for there to be an accepted symbol are rare, but that's history, not mathematics. Phi=(1+sqrt(5))/2. sqrt(2) is also irrational. Maybe I should call it upsilon. That's a rarely used Greek letter
. I could come up with most of representations on mathworld for my new irrational number. I could even come up with a magical sequence that converged to it. There'd be a couple of things I couldn't reproduce, but Upsilon would have some neat properties that Phi lacks also.
Most of the biological things people point to are really just things that are "about 1.5," or they have an obvious geometric connection as I said above. I could also say that sqrt(3)=1.732.... is a special number. Its close enough that most things would work out. I'll leave out the religious sarcasm about 3's
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