# Questions tagged [golden-ratio]

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

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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Golden ratio pattern in Sierpinski matrix eigenvalues

Stumbled onto the following observation. Defining a Sierpinski matrix recursively ...
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### 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 ...