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An interesting Number

I found this number quite a while ago just messing around with simple algebra and trigonometry. I am pretty sure it is too simple to have not been found before me, but it has some interesting qualities to it.

It is found by constructing the golden rectangle and drawing a line segment to BF and finding its length (this is not the arc illustrated in the image). If the number is 'b': \[b = (\sqrt{(5-\sqrt{5})/2}) \approx 1.17557\] There are some relationships between it and the golden ratio of course: \[b = \frac{\sqrt{(\varphi x-x)^{2}+x^{2}}}{x}\]

\[\Phi = \sqrt{(b^{2}-1)}\]

\[4(b^{2}-1)=(\frac{2}{\varphi})^{2}\] And there is one last relationship which is, at least to me, very interesting. We can represent the golden ratio as \(2cos(36^{\circ})\). And what is interesting is that, using simple math, you can get this relationship:

\(2\cos(36^\circ) = \frac{\sqrt{5}+1}{2}\)

\(\cos(36^\circ) = \frac{\sqrt{5}+1}{4}\)

\(\sin(36^\circ) = \frac{x}{4}\)

\(4\sin(36^\circ) = 2b\)


\(\boxed{\varphi^{2}+b^{2} = 4}\)

Which leads to an interesting question, which I cannot prove, but seems to be the case. It seems that 'b' is the only constant which gives a rational answer to the equation: \[\theta = \tan^{-1}\left ( \frac{\varphi}{\sqrt{n^{2}-\varphi^{2}}} \right )\] If theta is measured in degrees and \(n \geq 2\). It is also the smallest whole solution to this equation.

Anyway, that is the strange number; I hope it is interesting to someone.

Note by Drex Beckman
1 year ago

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I find this very interesting but I am unable to understand what is \(x\). Akshay Yadav · 1 year ago

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@Akshay Yadav Just the length of the square. You can just set them as one. For any real x, you should get the same number. Drex Beckman · 1 year ago

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