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

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Golden Ratio Approximation

$$\sqrt{1000}-30.0047 \approx \varphi $$ $$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$ Simplifying Above expression we get ...
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2answers
152 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
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1answer
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A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
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2answers
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Integral $\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx$

How can we evaluate this definite integral $$I=\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx,$$ where $\displaystyle\phi=\frac{1+\sqrt5}2$ is the golden ratio?
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3answers
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Is $e^{e^{2}}$ a relatively good approximation for $1000\phi$? [closed]

Yesterday night, I found that $e^{e^{2}} \sim 1000\phi$, where $\phi$ is the golden ratio. I believe that it is correct to four decimal places. Would it be considered a relatively good ...
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0answers
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Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
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1answer
73 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
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The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
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2answers
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proof by induction for golden ratio and fibonacci sequence

I have to prove the following equation by induction for $$x = \phi$$ I am stuck and I don't know how to proceed. This is the equation $$ \phi ^n = f_n\phi + f_{n-1} $$ where $f_n$ is the nth term ...
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Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

Let $\varphi=\frac{1+\sqrt5}2$ (the golden ratio). How can I simplify the following expression? $$7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$$
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1answer
59 views

How to find the center of a log spiral?

Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers ...
3
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1answer
72 views

Mathematical importance of the golden ratio [duplicate]

I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with ...
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4answers
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How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

How to compute the integral, $$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$ where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?
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1answer
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Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
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4answers
408 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??
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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*} ...
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1answer
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$\phi$, 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 ...
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0answers
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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 ...
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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. ...
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1answer
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$ab-(a+b)(a-b)=0$ 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?
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0answers
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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 $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
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2answers
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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 ...
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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 ...
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6answers
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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 ...
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2answers
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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 ...
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3answers
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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
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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) = ...
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2answers
544 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 ...
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0answers
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Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
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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: ...
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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 ...
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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 ...
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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+ ...
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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 ...
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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 ...
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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 ...
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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 + ...
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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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 ...
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1answer
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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!
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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 ...
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2answers
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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 ...
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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.
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0answers
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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 ...
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3answers
184 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 ...
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3answers
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$\pi$, $e$, $\phi$, 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 ...
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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 $\phi$ ...