- #1

nomadreid

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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?

[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?

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