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

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New Golden Ratio Construct: which one of my constructs is superior/simplest--squares & circles or just circles?

I have found yet another golden ratio construction. Geogebra gives it the value of 1.61803398874990 to the ratio between the yellow and blue lines in the figure below, which is the golden ratio PHI. ...
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1answer
27 views

Found a New Golden Ratio Construction with Equilateral Triangle, Square, and Circle. Geometric/Trigonmetric proof?

The below figure discloses a new golden ratio construction with an equilateral triangle, square, and circle. Geogebra gave me the value of the golden number 1.618 for the ratio of the yellow line to ...
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2answers
38 views

The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
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34 views

The Golden Ratio in a Circle, Triangle, and Square: simple geometry/trigonometry construction.

I believe that I have found the golden ratio in the below figure. Geogebra is saying it is very close. Might anyone have a geometric/trigonometric proof? An equilateral triangle ABC is inscribed in a ...
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1answer
35 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
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4answers
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Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square.

A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ...
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PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
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58 views

Is this well known? [duplicate]

How to prove $$1+\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\cdots}}}} = \left(\sqrt{5\phi}-\phi\right) e^{2\pi/5}.$$ i dont know how to do it. like if there were repeating ...
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Proofing that the Lucas numbers come closer to the Phi rounded numbers then the Fibonacci numbers.

Morning everyone, Bit of background, I'm a mid level programmer with very limited mathematics skills. As part of an assessment for a new role I've been asked to complete a technical task which ...
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1answer
61 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
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0answers
25 views

Golden Ratio Sequence

The golden ratio solves the equation: $$x^2-x-1=0$$ Equivalently: $x-(1/x)=1. $ What about generating a sequence of ratios $R_n$, by the equation: $x-(1/x)=n $ Has this been studied? We would ...
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2answers
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Prove that $\sum_{n=1}^\infty \left(\phi-\frac{F_{n+1}}{F_{n}}\right)=\frac{1}{\pi}$

So, I know that $$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$$ where $F_n$ stands for the n'th Fibonacci number I was interested in measuring the error of the convergence of the above limit and was ...
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1answer
88 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
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1answer
34 views

Weird informatic problem with Fibonacci numbers in which I have some troubles

I don't know what happended to this website but for months I am not able to connect me in it. As I understand it the website is closed. It is in this website I found this problem. Let $L$ be ...
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1answer
86 views

What is the math behind equal-spacing divider tool?

I want to know what is the math behind this tool, the 10 point divider (Full size image here). This tool is used to measure equidistant spaces and it's proportional, so you can scale it as much as ...
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1answer
54 views

Two ٍEquations Defining the Golden Ratio Differently

In most sources, the Golden Ratio has been referred to as: $\phi = \frac{\sqrt{5}+1}{2}$. Nevertheless, in the book " The Theory of Numbers, a Text and Source Book of Problems ", by Prof. Andrew Adler ...
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Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
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1answer
54 views

Golden ratio in goniometric function

I stumbled upon the function $$f(x)=\sin(x)(\sin(x)+2\cos(x)).$$ Now I noticed this function has maxima and minima at values $$\frac{1\pm\sqrt5}{2},$$ exactly the golden ratio. Computationally, this ...
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1answer
44 views

Golden ratio in regular pentagon

Let $ABCDE$ be a regular pentagon, F in arc small BA. Show that $\frac{FD}{FE+FC}=\frac{FB+FA}{FD}=0.618033$ the golden ratio; and $FD+FB+FA=FE+FC$
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Does there exist a derivative of the Golden Ratio (equation)?

On my calculus exam, there was a question asking "Is the Golden Ratio differentiable? If so, use the definition of a derivative to show it." The "definition of a derivative" is ...
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1answer
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Find conditions on $C$ and $C^{\prime}$ so that the spirals $r = Ce^{\varphi/a}$ and $r = C^{\prime}e^{\varphi/a}$ are the same

This question is related to one I asked here about the logarithmic spiral. In the linked problem, I had to find and sketch the image of the straight line $z=(1+ia)t+ib$, for $-\infty < t < ...
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Euler exponential continued fraction to compute the trigonometric functions and the golden ratio

Using the Euler continued fraction for the exponent, which is convergent everywhere on the complex plane: $$e^{-z}=1-\cfrac{z}{1+z-\cfrac{z}{2+z-\cfrac{2z}{3+z-\cfrac{3z}{4+z-\cdots}}}}$$ We can ...
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Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
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1answer
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Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
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Evaluating $\int_{0}^{\log\phi}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du$, when $\phi$ is the golden ratio

When I do $$e^u=x+\sqrt{x^2+1}$$ in (17) page 9, see here, after some computations then I obtain $$\frac{\zeta(3)}{10}=\int_{0}^{\log(\phi)}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du,$$ where ...
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1answer
67 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
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plot golden spiral along with rectangles showing the area split

I can see it's relatively simple to plot the golden spiral using the equation: $$r=e^{b\theta}$$ where $b={{ln(\varphi})\over{\pi\over2}}={{2ln({1+\sqrt5\over2}})\over{\pi}}=0.30634896253$ Here's ...
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Golden Ratio question

Question: A bar has a rectangular cross sectional area which has dimensions based on the Golden Ratio. A tension force of 500kN was applied on the side of this bar and the other side clamped. This ...
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Time Complexity recurrence

When we have the recurrence $T(N)=T(N-1)+T(N-2)$, one normally uses $x^N$ and solves for $x$ which gives the golden ratio. But why does one use $x^N$ and not something else, like $\log(N)$ or $N$?
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3answers
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Is there a pattern to the golden ratio number figures?

The golden ratio or phi is 1.6180339887498948482045... I am wondering if there is a pattern in the numbers so given a certain set of figures, you are able to figure out the rest of the figures ...
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A variation of Viète's formula — an infinite product of nested radicals [duplicate]

Viète's formula expresses an infinite product of nested radicals in terms of $\pi$. Let $$a_1=\sqrt2,\quad a_n=\sqrt{2+a_{n-1}}.$$ Note that $\lim\limits_{n\to\infty}a_n=2.$ Then ...
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1answer
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“Extending” the calculation of the golden ratio using square roots (not silver-ratio)

I'm looking at the following formula: $x =\frac{-n+\sqrt{n^{2}+4n}}{2}$ For $n=1$ this this gives $0.618...$ and then $\frac n x$ gives $1.618...$ which is $\phi$, the golden ratio. What ...
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1answer
86 views

Weakly unimodal function using Golden Section Search

I was going through the Golden Section Search https://en.wikipedia.org/wiki/Golden_section_search and as I understand it should work for every unimodal function. Here, the definition of unimodal ...
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1answer
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Summation of a series involving powers of Fibonacci numbers.

I'm interested in this series: $$\mathcal S_p=\sum_{n=1}^\infty\frac{\left(F_n\right)^p}{2^{np}},\quad p\in\mathbb N,\tag1$$ where $F_n$ are the Fibonacci numbers, defined by the recurrence ...
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1answer
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On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt ...
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4answers
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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 ...
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1answer
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Is the golden ratio a transcedental number?

I have been studying the concept of transcedental numbers. Till now, I had taken it for granted that all important numbers like pi and e were transcedental. I have no reason for assuming this or for ...
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1answer
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A different way to calculate number of primes less than a particular number x

According to Wikipedia, series representation of logarithmic integral function is as follows: Now, as per my findings from this we can also calculate number of primes less than a particular number x ...
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1answer
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Proof by induction that *p* = 1/*p*-1 in golden rectangle exercise

The initial rectangle's dimensions is L0 for the length and l0 for the width. A golden rectangle can be obtained when it has the same proportions as the initial rectangle, so p = L0/l0 I am first ...
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Is it meaningful in attempting to prove relation between prime numbers and Golden ratio? [closed]

I am trying to define a model which will give number of prime numbers less than or equal to a particular number $N$. I am using golden ratio $\phi$ in this model and I am quite satisfied with my ...
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2answers
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Relationship between golden ratio powers and Fibonacci series

Can anyone prove the following equation? ($F_n$ is the $n$th element of Fibonacci series and $n \in N$.) $\phi = 1 \times \phi + 0$ $\phi^2 = 1 \times \phi + 1 $ $\phi^3 = 2 \times \phi + 1 $ ...
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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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189 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
238 views

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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1answer
88 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 ...