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

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A BBP-type series

The BBP-type series $$ \frac{\pi}{2} \, \left( \frac{\alpha^{2}}{5} \right)^{\frac{1}{4}} = \sum_{n=0}^{\infty} \left[ \frac{1}{10 n + 1} + \frac{\alpha}{10 n + 3} - \frac{\alpha}{10 n + 7} - \...
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How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$

Here I mean the limit of the following sequence: $$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$ $$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$ $$p_3=\int_0^...
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Relating fibonnaci sequence, lucas numbers and golden ratio to make a research question?

I am planning to write a high school level maths essay of approximately 4000 words. I do find Fibonacci sequence, Lucas numbers and Golden ratio amazing and want to research further on them, the thing ...
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1answer
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Any more of this type? $\sum_{j=1}^{k}j\phi^j=\phi^{2k}$

Curious equations of phi $\phi={1+\sqrt5\over 2}$, golden ratio Here are two examples, $2\phi^2+1\phi^1=\phi^4$ $4\phi^4+3\phi^3+2\phi^2+1\phi^1=\phi^8$
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Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? [closed]

Below please enjoy a Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? The below construction is created by beginning ...
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The amazing golden ratio harmonies in a pentagon and decagon. Golden triangles naturally emerge, but do golden rectangles also emerge?

Consider a pentagon and decagon with sides of the same length drawn as shown in the diagram. Golden ratios will then abound, with the simplest perhaps being the ratio of a/b. Another remarkable ...
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1answer
312 views

A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?

Note: this construction is a vastly expanded version of my earlier construction here: Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or ...
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1answer
63 views

New Golden Ratio Conjecture with Triangle and Square: It is very close, but is it really the golden ratio?

Geogebra gives me 1.616 for the ratio of the blue segment p to the red segment q instead of the golden ratio 1.618 for the construction shown below, so it could be close to PHI, but no cigar. This ...
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3answers
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New very simple Golden Number Ratio PHI construction with Circle and Two Equal Segments of Circle Diameter. Is there prior art? Proofs?

Geogebra gives me the golden number PHI to fifteen decimal places for this simple construction illustrated below wherein the ratio of the blue line i to the red line h is PHI or 1.6180.... The golden ...
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New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
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Pythagorean triplet triangle associated with Golden Ratio Construction. [closed]

MoreGoldenPythagorusTriangle Just recognised simple connection by Astrophysics Math with a particular Pythagorean triplet proportion leads to an easy method to construct $ \varphi$ or its reciprocal. ...
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3answers
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New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art? [duplicate]

This question is different from a previously asked question (linked above) as this golden ratio construction only utilizes two circles and a line, and is thus far simpler than the golden ratio ...
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2answers
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New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?

The below Golden Ratio Construction results in a ratio of PHI (1.6180...) between the blue line and red line, as found in Geogebra. This seems like a simple construction of the golden ratio, yet so ...
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1answer
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Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly

Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio ...
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2answers
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Showing that $\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{2^n+1}L_{2^n+1}}\right)=\frac{3}{\phi^2}$

Infinite product $F_{n}:=[1,1,2,3,5,8,\cdots]$ and $L_{n}:=[1,3,4,7,\cdots]$ for $n=1,2,3,\cdots$ respectively. $\frac{1+\sqrt5}{2}=\phi$ Show that, $$\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{...
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2answers
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New Golden Ratio Construct with Geogebra using Square and Triangle with Same Base Width. Geometric proof of golden section?

The below construct of the golden ratio, based on the ratio of segment c to segemnt b, is so very close to PHI. Geogebra gives the value of 1.61957 instead of 1.61803. Might anyone have any insight ...
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1answer
76 views

What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Below I drew a golden rectangle in a pentagon in Adobe Illustrator. What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? ...
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1answer
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Golden Ratio & Fibonacci - Charles de Gaulle 13-unit two-beamed cross problem.

Here is the question: The two-beamed cross, made popular by Charles de Gaulle, is formed from 13 unit squares as shown below. A straight line $BC$ drawn through point $A$ divides the cross in such ...
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2answers
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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
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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
146 views

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

Geogebra gave me 1.618. . . . for the following Golden Ratio construction shown below. First off, has anyone seen anything similar to this construction? Basically begin with an equilateral ...
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1answer
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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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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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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
81 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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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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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
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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
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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
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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
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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
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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
57 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 $f'(x)=\displaystyle\...
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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 K}\limits_{i=1}^{2n}...
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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 $\phi=\frac{1+\...
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1answer
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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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Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$

I experimentally discovered the following conjectures: $$\Re\Big[1800\operatorname{Li}_3(i\,\phi)-24\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=100\ln^3\phi-47\,\pi^2\ln\phi-...
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1answer
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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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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 $$\prod_{n=1}^\...
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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
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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 ...