Physically relevant: fractals, phi?

[nomadreid]

There are two subjects which pop up a lot as having physical examples (or, more precisely, where their approximations have), but many (not all) of them seem rather indirect or forced. For example:

[1] phi (the Golden ratio) or 1/phi:

(a) trivia: sunflowers and pineapples giving the first few members of the Fibonacci series, which when taken to infinity gives ratios whose limit is phi
(b) forced: the ratio of an electron's magnetic moment to its spin angular momentum
(c) hypothetical: in models for Fibonacci anyons.
(d) indirect: phi is used to construct Penrose tilings, which are put into equivalence classes, upon which a groupoid C*-algebra is formed, upon which several other structures are formed, which gives a non-commutative algebra which resembles some aspects of quantum systems. Alternatively, the tilings are used as an example of the principles of 3D quasicrystals.

[2] Fractals:

(a) forced: the fractal dimension of galaxy distribution
(b) OK: applications of chaos theory
(c) self-similarity: hey, a straight line is self-similar, that doesn't make it a fractal.
(d)indirect: The Weierstrass function killing assumptions about differentiability being nowhere differentiable, or as or the Voronin universality theorem giving nice approximations to ... ah, wait, that's pure mathematics, which of course has indirect ties to physics, but...

So, are these two subjects really much directly of physical importance, or merely a couple of curiosities?

 

[sophiecentaur]

I don't know the answers to your question but it again raises the question of where Maths comes from and what it actually is. Even down to the question "What is the 'two-ness' of two bottles or two metres?" that makes us able to do similar calculations with them. Interestingly, Computer Coding has made us think about stuff like this when we very well not have found the need, before, to discuss types of variable.