Golden ratio discovered in quantum world (2010) | Hacker News Reader

Golden ratio discovered in quantum world (2010)

sciencedaily.com

20 pointsthomyorkie9 comments

Any second order recurrence equation (second order equations are very common tools for real world modelling) will have a golden ratio in it.

    f(n) = f(n-1) + f(n-2)

Regardless of the f(0) and f(1) you'll get a golden ratio.

This sounds similar to how, when one uses a wheel for measurement, the equations always have PI involved somehow.

The world has Fibonacci-like properties all over, not because we choose to view it through that lens, but because it's such a simple and ubiquitous quality of reality. In many cases it's easy to see how "the new value is the sum of the two previous values" is a concept that describes basic processes. You would be hard pressed to find a more direct description of those effects.

Exactly. The reason Newtonian physics is so successful is because almost anything can be described by a second order differential equation. The world behaves as simple as that and it would be quite weird if it required a larger order to describe it.

Yes but that works for this equation only because its characteristic polynomial is x²-x-1, whose roots are the golden ratio and its conjugate. For a general recurrence equation (order notwithstanding) you can get arbitrary asymptotics (dominated by the characteristic root of highest magnitude).

What I find interesting is that this class of sequence's ratio converges to the golden ratio as n approaches infinity, but the quantum system produced the golden ratio between its first two values.

This seems to suggest a more straight forward derivation (x + 1 = x^2) than infinite recurrence. [0]

[0] https://en.m.wikipedia.org/wiki/Golden_ratio#Calculation

If you want to solve a recurrence, you can do that by assuming the general form of the solution (which is c*a^n for the above), or transforming it into exactly that polynomial you listed, you'd get something similar by using generating functions too.

> second order equations are very common tools for real world modelling

... for example, in the discretization of the second derivative operator.

Never mind the golden ratio. The discovery of E8 in a physical system is badass.

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