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Comment by spooningtamarin | Hacker News Reader

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.

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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.

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