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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)

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.

stephengillie 11y ago
This sounds similar to how, when one uses a wheel for measurement, the equations always have PI involved somehow.
- Udo 11y ago
  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.
  - spooningtamarin 11y ago
    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.
deckar01 11y ago
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 https://en.m.wikipedia.org/wiki/Golden_ratio#Calculation
- spooningtamarin 11y ago
  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.
chestervonwinch 11y ago
> second order equations are very common tools for real world modelling ... for example, in the discretization of the second derivative operator.
an_to_nio 11y ago
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).

stephengillie 11y ago

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

Udo 11y ago
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.
- spooningtamarin 11y ago
  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.

deckar01 11y ago

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

spooningtamarin 11y ago
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.

chestervonwinch 11y ago

> second order equations are very common tools for real world modelling ... for example, in the discretization of the second derivative operator.

an_to_nio 11y ago

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).