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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Calculate the golden ratio of a song/time.

Let's say I had a song last for 300 seconds (5 minutes). How would I use the golden ratio (1.61803399) to find what point in the song the golden ratio is at (in seconds). An example: Uptown Funk is 270 seconds long, the "break down" of the song is…
AZ Games
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Possible Pythagorean relation with Golden Ratio $ \phi^2+e^2 \approx \pi^2$

While study Numerics and playing with famous constants ($e$, $\pi$, Golden ratio) I came across the following relation $$ \color{blue}{1.6^2+2.7^2 = 9.85\approx 3.14^2}$$ This is nothing special but patently, by first oder approximation one glimpses…
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Golden Ratio example not satisfying ratio

Golden ratio says $\frac{a}{b}=\frac{a+b}{a}$ and it shows up in rectangle also. If we take rectangle of $a=4$ and $b=2$. then $\frac{a}{b}=\frac{4}{2}=2$ and $\frac{a+b}{a}=\frac{4+2}{4}=\frac{6}{4}=1.5$. But $1.5 \ne 2$, how does this hold true?
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Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel?

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 with a 5x5 array of squares. A circle is then inscribed…
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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 construct is created by drawing a square and then…
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Why does this number appear?

In the Fibonacci Spiral, when you divide the area of the square, which holds that specific part of the spiral by the area of that specific section of the spiral, you get the root of the Golden ratio, ($\approx 1.2730$). Can anyone explain why this…
billy606
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Q&A and generalisation: On fractions and square roots

What is the value of $$\frac{294395}{294393}-\left(\frac{588789}{588787}\right)^2?$$ Motivation Have you ever tried pressing the square root button repeatedly on your scientific calculator to see how quickly (or slowly) a number reached $1$? If…
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