Mathematics Stack Exchange
Mathematics Stack Exchange

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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Solve for golden ratio value using recursion?

The golden ratio is defined as: $$\frac a b = \frac {a+b} a.$$ It comes down to: $$\varphi = 1 +\frac 1 \varphi.$$ Is there a way to solve for $\varphi$ computationally/recursively? Is there any other way to solve it?
Alexander Mills
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How to calculate phi with a random number generator?

I recently saw this as an interview question and was completely stumped by it.. Calculate phi using numpy.random.uniform I was thinking $$a + \frac{b}{a} = \frac{a}{b} $$ might be useful in that you could simply generate random numbers between 0 and…
Learning stats by example
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How does this drawing of a man's body in a pentagram suggest relationships to the golden ratio?

According to Wikipedia it does, but I can't see how. Wikipedia context: https://en.wikipedia.org/wiki/Golden_ratio#/media/File:Pentagram_and_human_body_(Agrippa).jpg
math wannabe
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Correct way for calculating Phi (golden ratio)

I've seen in some of the resources, that Phi is calculated in a way that the whole (=1) is divided into two parts: ---------------------x----------------------|---------1-x-------------- $\frac{1}{x}=\frac{x}{1-x}\\1-x=x^2…
derive
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How does Plato's description in the Timaeus relate to the golden ratio?

In the Timaeus, Plato states For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean - then…
user1301930
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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?
user1894220
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How does $2\pi(1-\frac{1}{\phi})$ become $\pi(3 - \sqrt 5)$, where $\phi$ is the Golden Ratio?

Help me please to understand how does the Golden Angle on this mathworld page is derived: https://mathworld.wolfram.com/GoldenAngle.html I can't understand how does it transformed from $2\pi(1-\dfrac{1}{\phi})$ to $\pi(3 - \sqrt 5)$. After…
Axel
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