Questions tagged [golden-ratio]

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

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Does anyone know the source of this theorem?

I'm writing a paper on $\beta$-expansions, or $q$-expansions, in non-integer bases and I cannot find a source for this theorem. Here $\gamma = \frac{1 + \sqrt{5}}{2}$. The theorem: (a). If $1 < \...
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108 views

$\sqrt{2} \ln \pi \approx 1.618033…$, the golden ratio. Why?

$\sqrt{2} \ln \pi = 1.618892$… is approximately equal to the golden ratio $\phi = 1.618033$… . Is this just a coincidence? Could it be some kind of first-order approximation?
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Which of the following numbers is a Fibonacci number; $(A) 75023$ $(B) 75024$ $(C) 75025$ $(D) 75026$?

This question appeared in one of the mathematical societies exams in Saudi Arabia. No calculator is allowed. The required time to solve one question is $4$ minutes (on average). There is only one ...
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An identity involving $\sum_{k\geq1}\frac{(-1)^{k-1}}{k^5 \binom{2k}{k}}$ and golden ratio

J. Borwein's book "Experiments in mathematics: computational paths to discovery", page 137 states that: $$\sum_{k\geq1}\frac{(-1)^{k-1}}{k^5 \binom{2k}{k}}=2\zeta(5)-\frac{4L^5}{3}+\frac{4\pi^2L^3}{9}+...
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How can I construct a Golden Ratio Gauge like this? What shall be the legs of such a gauge?

How can I construct such Golden Ratio Gauge? What shall be the lengths of respective legs (their parts)? $M, P, N$ - are sharpened tips (endpoints) $A, B, C, D$ - joints (screw-points)
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On the formula, $\pi = \frac 5\varphi\cdot\frac 2{\sqrt{2+\sqrt{2+\varphi}}}\cdot\frac 2{\sqrt{2+\sqrt{2+\sqrt{2+\varphi}}}}\cdots$

I found a formula on google images when I was looking at some formulas for $\pi$ just for the fun of it, and I came across one that really startled me, and was quite reminiscent of Viète's product. ...
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A conjecture concerning Fibonacci

First of all, I'm aware that this conjecture has probably already been made and proved. I was just playing around, and I'd like to know whether it is in fact true and perhaps a hint as to how to prove ...
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1answer
25 views

(why) is this ratio the golden ratio?

Looking at a slight variation of the fibonacci sequence f(x) = f(x-1) + f(x-2) + 1 where f(1) = 1, f(2) = 1 I'm trying to find the ratio of this sequence but can't figure out how. To get an ...
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37 views

Value of π*φ and π/φ

I found the expression of $\pi\varphi$ and $\pi/\varphi$ on Twitter via @AnecdotesMaths (https://twitter.com/AnecdotesMaths/status/1241032301783461890) in terms of an integral, and I wondered if ...
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How I can prove that the last digit of $1+6^n+2\times 3^n+7^n+4^n+3\times9^{n}+4\times8^n$ is $3$ or $9$?

I have checked the first $14$ digits of Golden ratio, and I have found some attractive properties. I have defined the sequence as $6^n+1^n+8^n+0^n+3^n+3^n+9^n+8^n+8^n+7^n+4^n+9^n+8^n+9^n$. Some ...
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Is it natural or the way that the numerical system was invented that these magics with numbers exist in Nature? [closed]

Why do these exhilarating magics with numbers exist in Nature? Is it natural or are they tricks of numbers the way they were invented or explored? Classical Fibonacci Sequence = $0, 1, 0+1=1, 1+1=2, ...
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Question Involving Golden Search Method and Fibonnaci Search Method theory

Hello, I am really struggling with this question. I (think) I have found the solution to the first constraint for a. Basically that the T(b-a) where 0 less than T less than 0.5 gives me the first ...
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27 views

Golden ratio of a circle and square

φ is the golden ratio. In the given figure how can we prove that perimeter of circle is not equal to perimeter of square. I am confused as it is not clearly described the figure. Then also how can we ...
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Show that: $\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$

How to show that? $$S=\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$$ Where $F_n$ Fibonacci number $F_n=\frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}$ $$F_n^2=\...
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65 views

How to show that : $4(-1)^nL_n^2+L_{4n}-L_n^4=2$

How can we prove that: $$4(-1)^nL_n^2+L_{4n}-L_n^4=2$$ Where $L_n$ is Lucas number We got $L_n=\phi^n+(-\phi)^{-n}$ $4(-1)^nL_n^2=8(-1)^n\phi^{2n}+8$ $L_{4n}=\phi^{4n}+(-\phi)^{-4n}$ $L_n^4=4\phi^...
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A question about the plastic number

The plastic number is well known to be the limiting ratio of the Padovan sequence (OEIS A000931), to wit, $$ P_n=P_{n-2}+P_{n-3}\\ \lim_{n\to \infty} \frac{P_{n+1}}{P_n}=p $$ However, it is also the ...
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37 views

Comparing $2$ infinite continued fractions

$A = 1 +\dfrac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}} \\ B = 2 +\dfrac{1}{2 + \frac{1}{2 + \frac{1}{\ddots}}}$ Given the two infinite continued fractions $A$ and $B$ above, which is larger, $2A$ ...
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Verifying a continued fraction related to $\logφ$.

The continued fraction is the following, $${1+\cfrac{1\cdot 2}{3φ+\cfrac{1\cdot 2}{5+\cfrac{3\cdot 4}{7φ+\cfrac{3\cdot 4}{9+\ddots}}}}}=\frac{2}{3\logφ}\tag{1}$$ Where, $$φ=\frac{1+\sqrt{5}}{2}$$ ...
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162 views

$\sum _{n=0}^{\infty} \frac{1}{(n+1) (n+2)} \left(\frac{1}{\lfloor n \phi \rfloor +2}+\frac{1}{\lfloor n \phi ^{-1} \rfloor +2}\right)$

How can we prove the following identity: $$\sum _{n=0}^{\infty} \frac{1}{(n+1) (n+2)} \left(\frac{1}{\lfloor n \phi \rfloor +2}+\frac{1}{\lfloor n \phi ^{-1} \rfloor +2}\right)=\frac{3}{4}$$ Here $\...
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Prove that if $a_{k+1} \leq C a_{k} a_{k-1}$ then $a_{k+1} \leq D a_k^\phi$.

Suppose we have a sequence $a_0, a_1, a_2, \dots$ of positive real numbers that satisfies $a_{k+1} \leq C a_{k} a_{k-1}$ for some constant $C$. If $a_0, a_1, C < 1$, then the sequence converges ...
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Approximation of $|\phi^{\pi}-\pi^{\phi}|$

Show that $$|\phi^{\pi}-\pi^{\phi}|\leq \operatorname{T},$$ where $\phi$ is the golden ratio and $\operatorname{T}$ the Tribonacci Constant Using a calculator, we have $|\phi^{\pi}-\pi^{\phi}|=1....
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68 views

Negative solution for continued fraction 1 + 1/(1+(1/… [duplicate]

I am interested in the continued fraction $$1 + \dfrac{1}{1 + \dfrac{1}{1 + ...}}$$ You can solve this by letting $$y = 1 + \dfrac{1}{1 + \dfrac{1}{1 + ...}}$$ Then since it is infinite $$y = 1 + ...
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Golden Angle Golden Spiral

Note. The answer must produce something self-similar that progresses in such a manner regardless of whether positive or negative values are graphed. Note. The "distance between 'arcs'" I'm talking ...
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Finding A Non-Arbitrary Version of A specific 'Cubic' Curve

Please See "Clarification" for more clarity I found a curve that I've been looking for; well sort of: The curve, shown in my fig., and Graphed Here, matches certain criteria: 1. Overall shape. 2. ...
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prove $ 1 < \varphi = \frac{1 + \sqrt{5}}{2} < 2$

Prove $ 1 < \varphi = \frac{1 + \sqrt{5}}{2} < 2$ Just want to see if my "reasoning is sound. 1) Showing $1 < \frac{1 + \sqrt{5}}{2}$ Consider $\frac{1}{2}$: $$\frac{1}{2} < \frac{1}{...
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Identifying An Unusual Curve (Parametric) [closed]

NOTE that: The upper boundary (for positive values of $t$) is defined in terms of infinity, that is at $t=∞$. (There is no lower boundary for negative values of $t$.) NOTE. To any re-reading this, I ...
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Hyperbolic Helix With Variable Pitch

I have tried to make this as easy to understand as I can. If you don't get what I mean, PLEASE, post a comment; thank you! I want to find parametric equations for a helix where the pitch grows in ...
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The following relation $\frac{a}{b} =\frac{b}{a+b}$ is satisfied for $a = 1.6b$ approximately. How to reach this result? [closed]

The following relation $\frac{a}{b}=\frac{b}{a+b}$ is satisfied for $a = 1.6 b$ approximately. My question is how to reach this result, I need a detailed explanation of each step of how $\frac{a}{b}=\...
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Converting a (2-D) Helix With Pitch Increasing at Multiples of an Angle to Hyperbolic Form

These equations $x,y=\sin(t), \frac{\phi^{\phi-3}}{\phi^\phi-1}\left(\phi^{\frac{\phi t}{2\pi}}-1\right)+1,$ produce a (2-D) helix where at every multiple of the golden angle ($2πφ^{-1}$) for $t$, the ...
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Parametric Equations for A 2-D Helix Where The Distance Between Loops are Powers of $φ$ at Multiples of The Golden Angle

I want to find parametric equations for a sine-wave ('2-D helix') of the form $x, y=\sin(t), f(t)$, where $f(t)$ causes the expression to: 1. Start at (0, 1) (for positive and negative numbers). 2. ...
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73 views

Is $\varphi$ the most irrational number? [duplicate]

A friend told me that the golden ratio constant $\varphi$, i.e., $\dfrac{1+\sqrt{5}}{2}$, is 'the most irrational number,' does anyone know if this is true and if so can it be proven? Thank you!
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Defining $α$ Via The Golden Angle in $\sin(t)·\left(α·φ^{t-{π/2}/π}+β-\frac{α}{φ^{1/2}}\right)^{-1}, \left(α·φ^{t-{π/2}/π}+β-\frac{α}{φ^{1/2}}\right)$

NOTE $0$ times the golden angle is a $G_1$ point, too, and should give a distance $φ^{-3}$. Some of this may be hard to visualize so, see my figs., also see a graph here. Also, see here for a ...
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A Sine Wave Where Alternate Distances Between 'Wave-center' Points Are Powers of φ

This may be hard to visualize without my graph, see here If $\phi=\left(\frac{1+5^{1/2}}{2}\right), \alpha=\phi^{-2}, \beta=1$, then the parametric equations, $$ (x, y)=\left(\sin(t)\cdot\left(\...
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189 views

Parametric Equations for A Logarithmic Sine-wave With Alternately Offset Points of Hyperbolic Tangency

I've been trying to derive the parametric equations for a specific type of sine-wave for quite some time, and now I think I know how to do it in principle but lack the skill in practice. So, I'd be ...
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Why does the derivative equation of the unit semicircle equation intersect the semicircle at $x=-\frac{1}{\phi}$?

I was playing around in desmos and I discovered something interesting, but I am nowhere near advanced enough to tackle this. The circle equation $f(x)=\sqrt{1-x^2}$ and its derivative (I don't know ...
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Calculating n'th Fibonacci number through golden ratio [duplicate]

Prove that $\frac{\phi^n}{\sqrt{5}}$ rounded off to the nearest integer gives the n'th Fibonacci number, where $\phi$ is the golden ratio.
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Deriving Parametric Equations For A Hyperbolic PHI Sine-Wave

This Question was formulated somewhat improperly: See here for the correct question: Parametric Equations for A Logarithmic Sine-wave With Alternately Offset Points of Hyperbolic Tangency !! Note. ...
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127 views

Solutions to $\prod_{n=1}^\infty \left ( 1+ \frac1{f(n)} \right ) = \varphi$

The constant $\varphi = \frac{1 + \sqrt5}{2}$ seems to show up everywhere. I am wondering what non-trivial function(s) $f$ satisfy $$\prod_{n=1}^\infty \left ( 1+ \frac1{f(n)} \right ) = \varphi$$ I ...
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71 views

Show that a diagonal in a pentagon is the golden ratio

I just learned that the diagonal of a pentagon (size 1) is the golden ratio (https://twitter.com/fermatslibrary/status/1210561047154872320) I tried to verify that, but ended up having to show that: $...
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Roots of $x^2+bx+c=0$ with $|b|\leq 1$ and $|c|\leq 1$ have absolute value at most the Golden Ratio. Coincidence?

I noticed a possibly interesting coincidence. Consider the set of quadratics $$ x^2 + bx + c $$ with $|b| \leq 1$ and $|c| \leq 1$. For all $x_0$ such that $$ x_0^2 + bx_0 + c = 0 $$ we have $$ |x_0| \...
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41 views

Golden Cuboid Equation [closed]

I am playing around with camper dimensions and I think it would be cool to have a camper as a "Golden Cuboid." I started with a golden rectangle that has the dimensions of 66.5 inches wide by 107.6 ...
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21 views

When is $ a+b \phi >0$?

What is an algebraic condition on rationals $a,b$ that characterises when $$ a+b \phi >0, $$ where $\phi=\frac{1+\sqrt{5}}{2}$.
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Proving $ \lim_{n\to\infty} \dfrac{ \Phi^{n+1} - (1 - \Phi)^{n+1}}{\Phi^{n} - (1 - \Phi)^n} = \Phi $

$ \Phi = \frac{1 + \sqrt{5}}{2} $ is the golden ratio I'm having hard time using proving that $$ \lim_{n\to\infty} \dfrac{ \Phi^{n+1} - (1 - \Phi)^{n+1}}{\Phi^{n} - (1 - \Phi)^n} = \Phi $$ dividing ...
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117 views

How can I prove analytically that the golden ratio is less then $\frac{\pi^2}{6}$.

Or stated in other terms, prove that $$\frac{1+\sqrt{5}}{2} < \sum_{n=1}^{\infty}\frac{1}{n^2}$$
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Golden ratio and floor function $\lfloor \phi ^2 n \rfloor - \lfloor \phi \lfloor \phi n \rfloor \rfloor = 1$ where $\phi = \frac{1+\sqrt{5}}{2}$

Prove that $\lfloor \phi ^2 n \rfloor - \lfloor \phi \lfloor \phi n \rfloor \rfloor = 1$ where $\phi = \frac{1+\sqrt{5}}{2}$ for all positive integers $n$ My thoughts: Tempted to take advantage of ...
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79 views

Gosper's identity for the golden ratio: $\frac{2^{2/5}\sqrt{5} \, \Gamma(1/5)^4}{\Gamma(1/10)^2 \,\Gamma(3/10)^2} = \phi$

Towards the end of a talk by Knuth (one of his Christmas talks, maybe the one from 2017), he mentioned in passing the following identity communicated to him by Bill Gosper (without proof, IIRC): $$\...
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Proof Phi is Irrational by using another Irrational Number

It is known to mathematicians that Phi (the golden ratio) is irrational number. The value of Phi is $\frac{(1+\sqrt5)}2$. The task is to use another irrational number (not $\sqrt5$) to proof the ...
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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} \...
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109 views

$2\arctan(\phi^{-n})=\arctan\frac{p}{q}$ or $\arctan\frac{p\sqrt{5}}{q}$, where $\phi$ is the Golden Ratio. Is there a pattern in the $\frac{p}{q}$s?

It is very interesting to know that $$\arctan\frac{1}{\phi} + \arctan\frac{1}{\phi^3}= \arctan 1 = \frac{\pi}{4}$$ where Golden ratio $\phi = \frac12(\sqrt5 +1)$ is in association with circle ...
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More Golden ratio - Link to $x^2 - y^2 =xy $

So, I like to mess around on Desmos. And there I am, looking at my all old graphs and I came across this one ($x^2-y^2=xy$) : And now that I know calculus (this was when I was $11$), I got all ...

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