Questions tagged [golden-ratio]

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

The golden ratio is defined to be the (unique) positive number $\varphi$ for which

$$\frac{\varphi + 1}{\varphi} = \frac{\varphi}{1}$$

or alternatively, the unique positive solution of

$$x^2 - x - 1 = 0$$

It can be written exactly as

$$\varphi = \frac{1 + \sqrt{5}}{2}$$

This number has been studied since antiquity, and the quantity frequently occurs in nature and art. It is also closely related to the Fibonacci numbers.

Reference: Golden ratio.

457 questions
41 answers

Is there any integral for the Golden Ratio?

I was wondering about important/famous mathematical constants, like $e$, $\pi$, $\gamma$, and obviously the golden ratio $\phi$. The first three ones are really well known, and there are lots of integrals and series whose results are simply those…
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Can the golden ratio accurately be expressed in terms of $e$ and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the infamous values, $\large\pi$ and $\large e$ The closest…
2 answers

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 of integers. What is meant by most irrational?…
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A golden ratio series from a comic book

The eighth installment of the Filipino comic series Kikomachine Komix features a peculiar series for the golden ratio in its cover: That is, $$\phi=\frac{13}{8}+\sum_{n=0}^\infty \frac{(-1)^{n+1}(2n+1)!}{(n+2)!n!4^{2n+3}}$$ How might this be…
2 answers

Prove that $\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $

I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0.61985841414477344973$$ Can we prove analytically…
Yuriy S
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On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as using $n = 5, 6, 7$ "ones" respectively,…
Tito Piezas III
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A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}.$$
5 answers

Why does this process map every fraction to the golden ratio?

Start with any positive fraction $\frac{a}{b}$. First add the denominator to the numerator: $$\frac{a}{b} \rightarrow \frac{a+b}{b}$$ Then add the (new) numerator to the denominator: $$\frac{a+b}{b} \rightarrow \frac{a+b}{a+2b}$$ So $\frac{2}{5}…
Joseph O'Rourke
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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?
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Show that the maximum value of this nested radical is $\phi-1$

I was experimenting on Desmos (as usual), in particular infinite recursions and series. Here is one that was of interest: What is the maximum value of $$F_\infty=\sqrt{\frac{x}{x+\sqrt{\dfrac{x^2}{x-\sqrt{\dfrac{x^3}{x+ \sqrt{…
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How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question: Are these numbers unique? How many other members are in the set if they exist?…
1 answer

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: But what I would like to know is: "Is the Fibonacci lattice the very best…
5 answers

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?
1 answer

A conjectured continued fraction for $\phi^\phi$

As I previously explained, I am a "hobbyist" mathematician (see here or here); I enjoy discovering continued fractions by using various algorithms I have been creating for several years. This morning, I got some special cases for a rather heavy…
2 answers

Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? ("Reciprocal addition" common for parallel resistors)

I have recently found some interesting properties of the function/operation: $x⊕y = \frac{1}{\frac{1}{x}+\frac{1}{y}} = \frac{xy}{x+y}$ where $x,y\ne0$. and similarly, its inverse operation: $x⊖y = \frac{1}{\frac{1}{x}-\frac{1}{y}} = \frac{xy}{y-x}…
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