Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$

learn more… | top users | synonyms

0
votes
0answers
20 views

Understanding golden section search

I don't understand it at all. The only thing I understand is that we have some interval in which we know a minimum lie and we know the function is unimodal. I also know that we are diving the interval ...
1
vote
4answers
176 views

Proof the golden ratio with the limit of Fibonacci sequence [duplicate]

Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$ The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$. Sorry as can be proceed??
10
votes
1answer
220 views

Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ …

Given two integer sequences \begin{equation*} \displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ \end{equation*} \begin{equation*} ...
0
votes
1answer
25 views

$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been ...
0
votes
0answers
21 views

Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...
3
votes
2answers
52 views

Fibonacci recursive algorithm yields interesting result

After writing a program in Java to generate Fibonacci numbers using a recursive algorithm, I noticed the time increase in each iteration is approximately $\Phi$ times greater than the previous. ...
1
vote
1answer
71 views

$ab-(a+b)(a-b)=0$ and the Golden ratio.

I have found: $b=a*\phi$ $b=a*(-\phi)$ $b=a/\phi$ Trying to find the correlation with the equation and phi, any insight how to demonstrate this or a proof?
1
vote
0answers
37 views

Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
1
vote
2answers
122 views

Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$. I know that these numbers add up, but is there any ...
0
votes
0answers
22 views

Do generalizations of the golden/silver ratios have geometric representations in corresponding polygons?

A silver mean of order n is $N_n=\frac{N+\sqrt{N^2+4}}{2}$. For N=1 we get $\phi$, which is found in a regular pentagon. For N=2 we get the silver ratio, which is found in a regular octagon. Will ...
20
votes
6answers
449 views

Why does every “fibonacci like” series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that ...
0
votes
2answers
116 views

Why is the radian golden angle $(1-1/\varphi)\cdot2\pi=\pi(3-\sqrt5)\approx2.39996$ so close to a 'nice' rational number?

I was reading about phytollaxis in plants and Fermat's spirals when I came across the Wikipedia article on golden angles. Surprisingly, the radian golden angle is very nearly approximated by a simple ...
4
votes
3answers
219 views

Inequality of the Fibonacci sequence and the golden ratio

How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. Any help is really appreciated
1
vote
2answers
54 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
4
votes
2answers
536 views

Why does this graph intercept both axes at the golden ratio?

Earlier, I was playing around with the Desmos Graphing Calculator, and I discovered that the following formula intercepts both the x and y axes at the golden ratio. I know that it makes sense, but I ...
5
votes
0answers
61 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
3
votes
2answers
91 views

How is the Binet's formula for Fibonacci reversed in order to find the index for a given Fibonacci number?

a question about the Fibonacci sequence: $$F_n =\frac{\phi^n-(-\frac{1}{\phi})^n}{\sqrt{5}}$$ This is the Binet's formula for the nth Fibonacci number. if I reverse it I can get: ...
1
vote
2answers
731 views

Relationship between Pi and Phi using the Great Pyramid of Giza?

In a documentation about the Great Pyramid of Giza, I heared following three theses about its measurements and the numbers $\pi$ and $\phi$ (the golden ratio). Measurement The Great Pyramid of ...
1
vote
0answers
70 views

Mean and Variance of Fibonacci Numbers

I would like to ask the community for feedback regarding the following two conjectures of mine: $\textbf{Conjecture 1}$ Let $\mathcal{F}_N^- = \{F_n:-N \leq n < 0\}$, i.e. be the set of Fibonacci ...
1
vote
1answer
96 views

Continued fraction of the golden ratio

It is known, that the continued fraction of $\phi = \frac{1+\sqrt{5}}{2}$ is $[\bar{1}]$. This can be shown via the equation $x^2-x-1=0$: $$ x^2-x-1=0 \Rightarrow x = 1+\frac{1}{x} = 1+ ...
0
votes
0answers
47 views

Golden Ratio Sandbox

This might be a little long so please bear with me. The Golden Ratio $\phi$ is defined as the single positive root of the polynomial $p(t) = t^2 - t - 1$. One can think of it as a line divided into ...
0
votes
0answers
64 views

Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
2
votes
0answers
82 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
0
votes
1answer
42 views

Sums of Golden Ratio Powers

I had a question regarding the following sum. Let $\phi$ be the golden ratio and $N$ be an even integer. \begin{array}{lcl} \sum_{n=1}^N (-\phi)^n & = & -\phi + (-\phi)^2 + (-\phi)^3 + ...
6
votes
0answers
107 views

Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
1
vote
1answer
81 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
3
votes
3answers
178 views

Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer.

How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio? This appeared in an answer to a question I asked previously, but I do not see how to prove ...
4
votes
2answers
75 views

Is there anything special about a graph with the golden ratio in its spectrum?

Given a simple connected graph $g$ with adjacency matrix $\mathbf{A}$. Let the spectrum $\lambda_1 < \lambda_2 < \ldots < \lambda_N$ be the eigenvalues of the equation $\mathbf{A} v=\lambda ...
2
votes
3answers
297 views

Simplifying the sum of powers of the golden ratio

I seem to have forgotten some fundamental algebra. I know that: $(\frac{1+\sqrt{5}}{2})^{k-2} + (\frac{1+\sqrt{5}}{2})^{k-1} = (\frac{1+\sqrt{5}}{2})^{k}$ But I don't remember how to show it ...
3
votes
1answer
176 views

Golden Ratio of Primes (Amateur)

Unable to find information elsewhere, so I'll try here. What two consecutive primes are closest to producing the Golden Ratio? Or two of any Primes? Has this been determined? Thanks!
9
votes
2answers
228 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
4
votes
2answers
100 views

Connection with golden ratio?

Consider the following problem: Let $p\in\mathbb{Z}[x]$ be a polynomial with integer coefficient. Suppose that the leading coefficient is 1, all roots are real and in $(0, 3)$. Find all ...
8
votes
1answer
107 views

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

Please help me to prove the identity $$_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0,$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
4
votes
0answers
97 views

Is there a golden pyramid?

Is there a golden pyramid? Or pyramides? Would they have some interesting properties (related to let's say packing, etc.)? Golden rectangle is said to be the most aestheticaly pleasing among ...
3
votes
3answers
162 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
6
votes
3answers
234 views

$\pi$, $e$, $\phi$, and sunflowers

While reading some internet materials on design, I came across this picture and comment: I found it a little bit surprising. I knew that the real sunflower follows golden ratio in some way (but I ...
2
votes
0answers
87 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
6
votes
2answers
1k views

What is golden ratio doing in this computer code?

In this file (related to random number generation), there is following line: private const int MSEED = 161803398; which resembles golden ratio. How come ...
9
votes
3answers
173 views

Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
1
vote
1answer
75 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
1
vote
1answer
96 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
3
votes
1answer
96 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
11
votes
5answers
439 views

What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?

Write $$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\vdots}}$$ so that $\phi_n=n+\frac{n}{\phi_n},$ which gives $\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$ We know $\phi_1=\phi$, the Golden Ratio, so ...
4
votes
2answers
398 views

Generalizations of the golden and silver ratios, and their significance

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios ...
3
votes
4answers
817 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
7
votes
4answers
745 views

Fibonacci numbers and golden ratio: $\Phi = \lim \sqrt[n]{F_n}$

Let $\Phi$ be the golden ratio and $F_n$ be the usual Fibonacci numbers. How can I derive the following formula? $$ \Phi = \lim_{n\rightarrow \infty} \sqrt[n]{F_n} $$ I know the usual relation $$ ...
1
vote
0answers
95 views

A sequence that converges to the golden ratio

This is elementary, but I found it somewhat surprising. Define $$ a_n = \frac{1+ a_{n-2}}{\sqrt{1+a_{n-1}}} \;,$$ where $a_1$ and $a_2$ are constants. For example, here is a plot for $a_1=5$ and ...
1
vote
5answers
277 views

Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]

We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}. $$ I wrote a script and found that this ...
0
votes
2answers
120 views

Prove that a Fibonacci number is greater than $ φ^n$

How can I prove the following: If $f_n$ is a number of the Fibonacci sequence and φ= $\frac{1+\sqrt{5}}2$, then $f_n > φ^n$ for every $n >2$? I have tried using induction but I can't ...
4
votes
2answers
733 views

Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]

Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove ...