Skip to main content

Questions tagged [golden-ratio]

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

Filter by
Sorted by
Tagged with
28 votes
8 answers
3k views

Where is the pentagon in the Fibonacci sequence?

It is common wisdom that "When you see $\pi$, there is a circle close at hand". For example: The periods of sine and cosine equal $2\pi$? Properly constructed, the right triangles that ...
No Name's user avatar
  • 1,055
4 votes
2 answers
115 views

Determine if a number is in the Fibonacci sequence using Binet's formula

I am wondering how to identify a Fibonacci number using Binet's formula. One of the approaches I tried was: Using Binet's formula, $$F_n=\frac{\varphi^{n}-(-1)^n\varphi^{-n}}{\sqrt{5}}$$ I multiply ...
Washington Fig's user avatar
0 votes
0 answers
10 views

conversion of 𝜑 vectors of regular icosahedron to unit vectors appear to be wrong.

I understand that common way of identifying the cartesian coordinates of an icosahedron is by using the coordinates, where 𝜑 is the golden ratio: (1+√5)/2) or cos(π/5.0) ...
Konchog's user avatar
  • 193
2 votes
2 answers
179 views

Why does dividing the nth Fibonacci sequence by the hypotenuse of a right triangle made from the sequence itself approach a constant?

Fn is the nth value of the Fibonacci Sequence given in the left column. I use the Fibonacci sequence as the legs of a right triangle and find the length of the hypotenuse of this right triangle for ...
QEntanglement's user avatar
0 votes
1 answer
69 views

What are some relations between the golden number or ratio $\phi$, and $\pi$?

What are some relations between the golden number or ratio $\phi$, and $\pi$? For example, by considering this answer https://math.stackexchange.com/a/744196/ ; by Steve Lewis. Now taking the point at ...
g.a.l.l.e.t.a's user avatar
0 votes
0 answers
28 views

What type of spiral is that on the picture ? and what is the formula of such?

I have found some types of spirals, and when I analysed those I have found, they do not met the criteria to shape the draw desired. And a observation point, bacause I think spirograph its a wrong name ...
FrakTool's user avatar
2 votes
2 answers
231 views

Golden ratio pattern in Sierpinski matrix eigenvalues

Stumbled onto the following observation. Defining a Sierpinski matrix recursively ...
vallev's user avatar
  • 396
3 votes
1 answer
68 views

Closed expression of a specific splitting of a continued fraction

Recently in my research I stumbled upon this splitting of a periodic continued fraction. I wondered whether there is any closed expression or literature on this topic. Visualizing continued fractions ...
Bindajoba's user avatar
4 votes
1 answer
129 views

Finding functions $h$ satisfying $h(h(n))+h(n+1) = n+2$ for all natural $n$. Why does the Golden Ratio appear?

Find all functions $h$, on natural numbers satisfying the functional relation, $$h(h(n))+h(n+1) = n+2$$ for $n$ being a natural number. Answer: So, the function is $h(n) = [n\alpha]-n+1$ where $\...
Krave37's user avatar
  • 543
4 votes
1 answer
188 views

Deciphering Salvador Dali's geometric formula $\frac{R}{2} \sqrt{n-2 \sqrt{5}}$ in "Leda Atomica"

Salvador Dali applied in his 1947–1949 painting "Leda Atomica" a concept that relates to the golden ratio or its conjugate. It is well known that Dali used mathematical formulas to elaborate ...
al-Hwarizmi's user avatar
  • 4,310
-1 votes
2 answers
137 views

Why is a=43 an exception?

There is a sequence. The first integer is positive, the second integer is negative. An alternating part is a part that switches between positive and negative (0 is not included) (a) I have a complete ...
confused's user avatar
25 votes
7 answers
1k views

Show by hand : $e^{e^2}>1000\phi$

Problem: Show by hand without any computer assistance: $$e^{e^2}>1000\phi,$$ where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$. I come across this limit showing: $$\...
Ranger-of-trente-deux-glands's user avatar
3 votes
0 answers
94 views

Why are there no identities for the golden ratio?

Why are there no identities for the golden ratio? While there are a plethora of identities for Fibonacci and Lucas numbers [see, for example, here and here: S Vajda, Fibonacci and Lucas numbers, and ...
Cye Waldman's user avatar
  • 7,768
1 vote
0 answers
116 views

Is the Imaginary Golden Ratio Unique?

The concept of the imaginary golden ratio is fairly well known and it is described here and has appeared in an MSE question here Imaginary Golden Ratio. The point of the imaginary golden ratio is that ...
Cye Waldman's user avatar
  • 7,768
3 votes
0 answers
172 views

The Golden ellipse

About a year ago I discovered the golden ellipse, which has many interesting properties It is an ellipse, the ratio between its large diameter and the distance between the two foci is equal to $φ$ $φ=\...
زكريا حسناوي's user avatar
1 vote
1 answer
44 views

Ratio of Fibonacci numbers

I have observed the following property: If $F_n$ denotes the $n^{th}$ Fibonacci number and $F_1=1; F_2=1$ Then $\frac{F_{2n+1}}{F_{2n}}>\phi$ And $\frac{F_{2n}}{F_{2n-1}}<\phi$ For all natural ...
Aarush Saharan's user avatar
0 votes
2 answers
138 views

Can the Fibonacci Spiral be expressed as a polar equation?

Related to this question: can the Fibonacci Spiral be expressed as a polar equation? I know the Golden Spiral can be and the Fibonacci differs in that it uses consecutive arcs of a circle for each ...
Nick's user avatar
  • 1,081
0 votes
1 answer
230 views

Fibonacci Spiral vs Golden Spiral?

I watched this video on constructing the Fibonacci Spiral. Does it differ from the Golden Spiral? The Fibonacci Spiral is constructed by using arcs of a circle on consecutive squares where the ...
Nick's user avatar
  • 1,081
0 votes
0 answers
53 views

Maximum absolute value of sum of m n-th roots of unity

Let $\omega, \omega^2,\dots,\omega^{n-1},1$ be $n$-th roots of unity, and consider $$S_n(m):=\max_{1 \le i_1 < i_2 <\dots<i_m \le n}|\omega^{i_1}+\dots+\omega^{i_m}|$$ i.e. the maximal ...
Peng Hao's user avatar
  • 153
0 votes
0 answers
83 views

How is the Gamma function related to the Golden Ratio?

I was researching the Gamma function and came across this in Wikipedia: The positive solution to Γ(z − 1) = Γ(z + 1) is z = φ ≈ +1.618, the golden ratio, and the common value is Γ(φ − 1) = Γ(φ + 1) = ...
Chuck's user avatar
  • 263
1 vote
0 answers
114 views

Triquetra proportions and Golden Mean?

I've been tinkering with drawings of Triquetras (triangular figures composed of three overlapping arcs), and wondering why this ancient symbol has appeared in so many cultures over so many ages. It ...
elvexo's user avatar
  • 31
1 vote
1 answer
255 views

Ways to "milk" the integral $\int^{1}_{0}\ln\left(\phi^{x}-\phi^{-x}\right)\ln\left(\phi\right)dx=-\frac{\pi^2}{20}$

After reading the MSE post on "Integral Milking", my first instinct was to try it out on one of my favorite integrals: $$\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$$...
Dylan Levine's user avatar
  • 1,728
0 votes
1 answer
114 views

Help with a solution this Fibonacci triangle question [closed]

I am currently studying maths at a high school level, I have been given this problem but can’t find any route into it, I have experimented with using Heron’s formula and the expression for the nth ...
RotterAlo's user avatar
1 vote
2 answers
115 views

Exponential Fibonacci and its recurrence-relation to ϕ. [closed]

1,1,2,3,5,8,... = Additive Fibonacci: a(n-1) * phi is asymptotic to a(n) 2,2,4,8,32,256,... = Multiplicative Fibonacci a(n)=a(n-1)*a(n-2): a(n-1) ^ phi is asymptotic to a(n) 2,2,4,16,65536,1.158*10^77....
Peter Woodward's user avatar
4 votes
1 answer
177 views

Is there a clever way to show that $\sin (\phi+1)<\frac12$?

I noticed that $\phi+1\approx 1.000015\left(\frac{5\pi}{6}\right)$, where $\phi=\frac{1+\sqrt5}{2}$, the golden ratio. So I wonder, is there a clever way to show that $\sin (\phi+1)<\frac12$, ...
Dan's user avatar
  • 25.7k
1 vote
1 answer
82 views

(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 ...
Ibsy's user avatar
  • 11
11 votes
1 answer
254 views

A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$

I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
Dylan Levine's user avatar
  • 1,728
-2 votes
2 answers
212 views

A conjecture for the golden ration via continued fraction $\phi=\frac{1+\sqrt{5}}{2}$

Playing with my own question Got a factored version of the Taylor's series? I found that : Define : $$f(x)=\frac{1}{\sqrt{x-2/x}}$$ Then it seems we have : $$\phi=f(f(\cdots f(\phi)\cdots))$$ ...
Ranger-of-trente-deux-glands's user avatar
0 votes
2 answers
234 views

Why is the continued fraction [1,1,1,1,1...] equal to ϕ?

I wanted to know what the continued fraction $1 + \frac 1{1 + \frac 1{1+ \frac 1{1+\frac 1{1+ \frac 1{1 + \ldots}}}}}$ would euqal to, so I chose $x=1+\frac1{x}$ because you can put the euqation in ...
Ziro's user avatar
  • 1
0 votes
1 answer
80 views

Explain this proof of Binet's formula?

I am confused about the proof offered here: https://sicp-solutions.net/post/sicp-solution-exercise-1-13/. The proof starts at Fib(n) = Fib(n - 1) + Fib(n - 2), which is true by definition. Then it ...
Noah J's user avatar
  • 115
3 votes
1 answer
99 views

Is there a graph that satisfies the golden ratio polynomial?

Is there a graph $G$ containing a bridge-edge $e$, such that if you delete the edge $e$, the resulting graph $G-e$ is isomorphic to $G\times G$? Such a graph, if it exists, would be a graph analogue ...
user326210's user avatar
  • 17.7k
2 votes
1 answer
65 views

Is my reasoning using limits correct for why 𝜙=1+1/𝜙?

let $$\lim_{n \to \infty}\frac{f_{n+1}}{f_n} = \phi$$ by definition, where $$f_n$$ is the nth fibonacci number then $$\implies \lim_{n\to \infty}1+\frac{f_{n-1}}{f_n}= \phi $$ $$\implies 1+\lim_{n\to \...
MonsterRamen's user avatar
4 votes
0 answers
85 views

Which $x \in \mathbb R$ satisfy a property related to sequences

This is a follow up to If $\sum\limits_{n=1}^{\infty}e_n x^n = 0$ always implies $\sum\limits_{n=1}^{\infty}e_n a_n = 0$, then $(a_n) = (x^n)$?. For $x \in \mathbb R$ we define the property $P(x)$ as ...
gerw's user avatar
  • 31.7k
2 votes
1 answer
89 views

Mixing Continued fraction and nested radicals does $D=-\pi+\ln(16)+(7\ln\pi)/2-7/3\arctan(\pi)$?

Let : $$B=\cdot\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdots}}}}}}}}$$ Let : $$D=\frac{1}{B+\frac{1}{B+\frac{1}{B+\frac{1}{B+\cdots}}}}$$ Then I conjecture with WA that ...
Ranger-of-trente-deux-glands's user avatar
4 votes
6 answers
298 views

How to solve $\sin\left(1+\frac{1+\sqrt{5}}{2}\right)<1/2$?

Problem : Find a geometric construction or a proof by hand to show : $$\sin\left(1+\frac{1+\sqrt{5}}{2}\right)<1/2$$ As attempt I introduce the inequality : $$\sin\left(1+\frac{1+\sqrt{5}}{2}\right)...
Ranger-of-trente-deux-glands's user avatar
11 votes
6 answers
395 views

Any cool ways to solve $\int_{0}^{\frac{\pi}{2}}\log(1+4\sin^2x)\,dx$

As per the title, I have solved the following integral $$\int_{0}^{\frac{\pi}{2}}\log(1+4\sin^2x)\,dx$$ I would love to see any insights, and solution processes anyone may have in solving it as well. ...
Person's user avatar
  • 1,123
6 votes
3 answers
2k views

The golden ratio in a parabola

It is nice that the golden ratio appears automatically when we are not looking for it. This is what happened to me when I was using GeoGebra and trying to solve a different problem that occurred to me:...
زكريا حسناوي's user avatar
-1 votes
2 answers
831 views

What are the known methods of drawing a golden rectangle with a ruler and compass?

Googling and Wikipedia, gives only the following construction A golden rectangle can be constructed with only a straightedge and compass in four simple steps: Draw a square. Draw a line from the ...
C.F.G's user avatar
  • 8,571
0 votes
0 answers
122 views

Proof that the sequence $x_n=f_{n+1}/f_n$ convergences to the golden ration $\Phi$

I'm supposed to proof that the sequence $x_n:=f_{n+1}/f_n,n\in\mathbb{N}$ converges to the golden ratio $\Phi$ where $f_n:=f_{n-1}+f_{n-2}$ is the Fibonacci sequence with $f_1=1$ and $f_2=1$. At first ...
Fynn Zentner's user avatar
0 votes
1 answer
99 views

Nested radicals $\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$ then $\lim_{x\to\infty}(g(x+1)-g(x))=^?1$

Well let the problem first : Conjecture : Let $x>M>0$, $b=\sqrt{x}$,$0<C<1$ then define : $$\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$$ Then it seems we have :...
Ranger-of-trente-deux-glands's user avatar
0 votes
2 answers
83 views

How to construct (with straightedge and compass) this diagram with four circles and two chords?

Inspired by this Sangaku-style question about a constellation of circles, I've come up with the following question. How can we construct, with straightedge and compass, the following diagram? ...
Dan's user avatar
  • 25.7k
11 votes
2 answers
541 views

Self-made Sangaku-style geometry problem involving chords and inscribed circles

In the diagram, circles (or disks, if you like) of the same color have the same radius. (For an explicit description of the diagram, see below.) Let $g=$ radius of the green circles, $r=$ radius of ...
Dan's user avatar
  • 25.7k
1 vote
4 answers
136 views

Solving for $x$ in $x^5+x^4+x^3+x^2+x+1=0$

So I was scrolling through the homepage of Youtube to see if there were any math equations that I thought that I might be able to solve when I came across this video by Blackpenredpen which was a ...
CrSb0001's user avatar
  • 2,652
1 vote
0 answers
55 views

Limit of a series of a side of the Golden Ratio

I try to find a limit of a series of a side from the rectangle of a golden ratio (Golden Cut, Goldener Schnitt). Just for fun to learn more about series. After hours of thinking I could need some help ...
Seminom's user avatar
  • 53
1 vote
1 answer
86 views

Golden number, diagonals of polygons and continued fractions

It is well know that, for a pentagon with unit side, the diagonal $\delta$ is such that $$ \delta : 1=1:(\delta-1) $$ so that its length is the positive solution of the equation $x^2-x-1=0$. i.e. the ...
Emilio Novati's user avatar
1 vote
1 answer
73 views

$\lim_{n\to \infty}\frac{F_{n+1}^{k}}{F_{n}^{k}}=\lim_{n\to \infty}\frac{F_{n+k}}{F_n}=\phi^{k},$ true? for odd $k$'s?

I was working with continued fractions and the Fibonacci sequence. And concluded this. Firstly, we know $$\lim_{n \to \infty}\frac{F_{n+1}}{F_{n}}= \phi$$ and easily you could prove it is using its ...
Mina Basilious's user avatar
2 votes
0 answers
80 views

Different way of proving: $\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)$

$$\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)\tag1$$ $\phi=\frac{1+\sqrt{5}}{2}$ $H_n=\sum_{k=1}^{n}\frac{1}{k}$ we expanded $(1)$ ...
Sibawayh's user avatar
  • 1,343
4 votes
7 answers
344 views

If $a^2-a-1=0$ where $a\gt0$, then what does $a^6$ equal? (Olympiad question)

$\color{white}{\require{cancel}{3}}$ So I was looking on Youtube for math equations that I thought that I could probably solve when I came across this video by the channel Maths and many more. The ...
CrSb0001's user avatar
  • 2,652
4 votes
3 answers
265 views

Why does there seem to be a relation between the fifth roots of $32$ and $\varphi$ (the golden ratio)?

I have the following problem on one of my assignments: find all five fifth roots of 32. What I find interesting is how $\varphi$ is related to the real component of most of the solutions. My ...
bwootton's user avatar
  • 125
-1 votes
1 answer
100 views

Show an integral inequality $\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$

Denote by $\phi$ the golden ratio then show the inequality : $$\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$$ Where : $$f\left(x\right)=\left(\frac{1}{x+1}\right)^{\frac{...
Ranger-of-trente-deux-glands's user avatar

1
2 3 4 5
11