Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$
26
votes
2answers
439 views
Why is $\varphi$ called “the most irrational number”?
I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
2
votes
4answers
104 views
What real number is exactly one less than its cube?
And does it have any of the special properties that the golden ratio (one less than its square) has?
1
vote
1answer
39 views
Can rainbows be considered to have mathematical patterns?
Of course it is evident that there is physics and mathematics involved in a rainbow, but my question is (as the title suggests) are there patterns to be found in rainbows? By patterns I mean ...
2
votes
2answers
91 views
How can I find an inverse to the Binet formula?
I'm already aware of the Binet formula $F_n = \frac{\varphi^n + \frac{1}{\varphi^n}}{\sqrt{5}}$. I'm attempting to find the inverse of that formula so I can find the position in the sequence of ...
5
votes
5answers
141 views
Continued Fraction [1,1,1,…]
If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
2
votes
2answers
149 views
understanding the Golden ratio intuitively
i am very interested in Golden Ratio and its value. the Golden Ratio itself is not hard thing to visualize and understand in 5 minutes. But i am trying to reach the historical, logical reasons of ...
2
votes
2answers
41 views
Non-isomorphic combinatorial classes with growth rate equal to the golden ratio?
In a couple of weeks, I'm giving a talk about the growth rates of some combinatorial classes.
In the introduction, I thought I'd present the class of tilings of a $n\!\times\!2$ strip with dominos, ...
4
votes
3answers
141 views
Why doesn't this induction “proof ”show $f_n = (\phi)^n + (1-\phi)^n$?
Here, $\phi$ is the golden ratio and $f_n$ is the $n^{th}$ Fibonacci number. The formula I'm using is actually the closed form of the Lucas numbers.
Let $n = 1$. Then $f_n = 1$ and $\phi + 1 - \phi ...
0
votes
2answers
47 views
Computing $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$
I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$.
But honestly I'm not even sure where to start. ...
4
votes
2answers
168 views
Prove that the sequence $x_{n+1} = {\frac{x_n+1}{x_n+2}}, x_0 = 1$ converges to $\frac{1}{\varphi}$
I've established that $(x_n)$ is monotonically decreasing, but don't know if I should attempt to show that $\frac{1}{\varphi}$ is the infimum of $(x_n)$ or use another method.
1
vote
1answer
91 views
Need formula for sequence related to Lucas/Fibonacci numbers
I am trying to get a formula for the nth term of the following sequence:
2, 14, 22, 58, 266, 398, 1042, 4774, 7142, 18698, 85666, 128158, 335522,...
It's not in OEIS and as far as I can tell ...
9
votes
3answers
436 views
Is the golden ratio overrated?
While reading many sources and watching many video's about nature and physics I wonder if the golden ratio is really that often occurring in nature and physics ?
If I look on the Wikipedia pages I see ...
2
votes
3answers
87 views
Does this number have an expression as square root or log of something
If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped.
$$1.52360679774998$$
The continued fraction calculator gives $1, 1, 1, ...
0
votes
0answers
58 views
Finding the kth n-anacci number
At this wolfram link the formula for the kth n-anacci number is given:
http://mathworld.wolfram.com/Fibonaccin-StepNumber.html#eqn8
(Eq. 4)
Not sure if I understand correctly. If I want the fifth ...
2
votes
1answer
72 views
Is it possible to extend phi to higher order polynomials?
Below, $x=\phi$ when $n=2$:
$$x^n-\sum_{i=1}^{n}x^{n-i}=0$$
($\phi$ being the golden ratio)
Is there a way to express $x$ in terms of $\phi$ for $n>2$?
4
votes
2answers
277 views
Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?
I have been looking at concise ways to represent irrational numbers using only integers.
I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...
0
votes
2answers
163 views
Solving Fibonaccis Term Using Golden Ratio Convergance
While solving this problem, I discovered that there is a relationship between the Fibonacci sequence and the golden ratio. After I got the correct answer via brute force, I discovered this ...
2
votes
0answers
259 views
Does this ratio converge to the Golden Ratio?
This infinite sequence a(n) starting: $1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 30, 45, 68, 104, 161...$ is the antidiagonal sums of a triangle that has several properties in common with the Pascal ...
3
votes
1answer
93 views
Golden parallelepiped
Define a golden parallelepiped as a $d$-dimensional box with side lengths
$(1, \phi, \phi^2, \ldots, \phi^{d-1})$, where $\phi$ is the golden ratio:
...
3
votes
1answer
259 views
Ulam's Spiral, Oppermann's diagonals and the Golden Ratio
With the help of our friends over here:
http://mathematica.stackexchange.com/questions/6219/ulams-spiral-with-oppermans-diagonals-quarter-squares
We created Ulam's Spiral with Oppermann's diagonals
...
17
votes
2answers
444 views
Approximation for $\pi$
I just stumbled upon
$$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$
which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I ...
4
votes
3answers
213 views
How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
How would one prove that
$$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$
where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?
