# Tagged Questions

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

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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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### $\phi$, 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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### 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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### $ab-(a+b)(a-b)=0$ 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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### 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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### 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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### 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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### 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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### 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
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 ($\approx1.61803399\ldots$) $$Fib(z) = ... 2answers 555 views ### Why does this graph intercept both axes at the golden ratio? Earlier, I was playing around with the Desmos Graphing Calculator, and I discovered that the following formula intercepts both the x and y axes at the golden ratio. I know that it makes sense, but I ... 0answers 83 views ### Why does the tribonacci constant have a trilogarithm ladder? When I came across the dilogarithm ladders of Coxeter and Landen, namely,$$\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$$... 2answers 145 views ### How is the Binet's formula for Fibonacci reversed in order to find the index for a given Fibonacci number? a question about the Fibonacci sequence:$$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: ...
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 ...