Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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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 this..
user1001001
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Golden ratio mod 1 distribution

If you plot the sequence $$x \leftarrow (x + \varphi)\ \mathrm{mod}\ 1$$ you get a nice scattering of numbers where no number is close to any previous number. This image shows the sequence after each iteration. You can (sort of) see that at every…
Timmmm
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How to prove $\phi=1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_nF_{n+1}}$

How do you prove $$\phi=1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_nF_{n+1}}$$ where $\phi$ is the golden ratio and $F_n$ is the $n$th Fibonacci number? I am aware that $\lim\limits_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$ and…
user373679
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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?
Lee Sleek
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Why do these two graphs intersect at 0.618 (golden ratio)?

I was playing around with Desmos Graphing Calculator and found that the two graphs below intersect at (-1,0), (0,-1), (1.618, 1.618), and (-0.618, -0.618). The latter two points are the golden ratio and the negative of the golden ratio plus 1. …
Tim K
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Golden spiral created using golden rectangles vs pentagram

I am trying to create a graphic that shows the golden spiral created using a pentagram and the golden triangles contained therein. I have drawn out the pentagram and golden triangles and the subsequent smaller triangles and pentagrams that are…
Benjam
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Confusion regarding Golden Ratio.

I read this post (Adi Dani) he wrote $\phi=\dfrac{-1+\sqrt{5}}{2}$ but Wikipedia shows that there is $+1$ not $-1$ involved in Numerator, please clear this confusion of mine that who is correct or it can be written in that way too.
mathlover
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(answered) found the golden ratio in something out of pure luck and want to know if there is any reason?

As a background I know nothing about how the golden ratio is used in actual mathematics or any formulae and such (only seen it used in a few examples I've seen online). But then while messing on the Desmos graphing calculator with just graph…
Ibsy
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lengths of sides in golden ratio isosceles triangles

The figure below shows three different isosceles triangles. every triangle is either 36-36-108 or 36-72-72. The base of the outermost triangle has length $\phi$. Find the lengths of both lines AT and MT. Can someone please help me figure this out?…
user477465
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Why does the golden ratio satisfy $\phi = 1 + \frac{1}{\phi}?$

I want to show this result starting from the property that $\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}$. I presume that to get $\phi = \frac{1}{\phi} $ I need to find the limit of the equation above. However, I'm unsure how to do this.…
mathstack
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Why isn't the golden ratio defined as the points where $f(x)=x^3-x^2-x$ are zero?

I was messing around on desmos, and when I plugged in $f(x) = x^2 - x - 1$, I get two points where $f(x)$ is zero, which are answers to the golden ratio. Why is this not used in the definition? It seems so much clearer to me. Link:…
user406613
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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 rational number, $2.4$. In the spirit of…
obataku
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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 + \ldots + (-\phi)^N \\ & = & (\phi^2 - \phi) + (\phi^4 - \phi^3)…
David Seres
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How does this proof of Fibonacci work

\begin{eqnarray*} F_{i+1}&=&F_{i} +…
GivenPie
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Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$?

Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$? Layman is trying to reversing golden ratio. Tell a real solution and complex solution please. Intuition says there should be a real value of $x$.
user104657
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