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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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 triangle. Inscribe a circle inside of it. Connect the…
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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 mirrors the title of this question. The programming…
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Prove $\phi^n = \phi F_n + \text{(another Fibonacci number)}$ using mathematical induction.

I need to prove the following equation using mathematical induction and using the phi values if necessary. $\phi^n = \phi F_n + \text{(another Fibonacci number)}$ In this proof, it is kind of hard for me to prove this with the part "another…
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Solving Fibonaccis Term Using Golden Ratio ConvergEnce

While solving this problem, I discovered that there is a relationship between the Fibonacci sequence and the golden ratio. After I got the correct answer via brute force, I discovered this relationship. One of the posters said this: The nth…
kakridge
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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 asked to make relations between Ln+1 and ln, and then…
Dhazard
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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 $ $\phi^4 = 3 \times \phi + 2 $ $\phi^5 = 5 \times \phi + 3…
user268388
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$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been found, but I thought that it was a cool discovery. My…
Taylor
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Can't find ANY golden ratio in the schroder house...

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to be using the golden ratio, but, when measuring, I…
binoculars
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Relationship between Pi and Phi using the Great Pyramid of Giza?

In a documentation about the Great Pyramid of Giza, I heared following three theses about its measurements and the numbers $\pi$ and $\phi$ (the golden ratio). Measurement The Great Pyramid of Giza had originally following measurements: $$s =…
Daniel Marschall
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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 properties I have noted are the following: 1/-The…
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Identifying An Unusual Curve (Parametric)

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 was tired when I wrote it and mixed up the…
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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 irrationality of Phi.
Monolica
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Golden Mean of Rectangle

ABCD is a rectangle with length and breadth in the ratio α : 1. It is divided into a square APQD and a second rectangle PBCQ, as shown. Show that the length and breadth of rectangle PBCQ are also in the ratio α : 1. Also this question could…
j.a456
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Algorithm to find floors of multiples of the golden ratio

What is an algorithm to calculate $\lfloor n\phi \rfloor$ given some integer $n$, where $\phi$ is the golden ratio? I am thinking the easiest way will involve calculating multiples of its continued fraction representation, since the golden ratio has…
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Powers of the golden ratio

Let $\phi$ be the golden ratio. I'm tasked to prove by other means than induction that $x$ in the next equation $$\phi^n =\phi F_n +x,$$ is actually a Fibonacci number. I have tried to apply Binet's formula to $\phi^n -\phi F_n$: \begin{align}…
Nilp Amin
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