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While looking at the Wikipedia entry for the Supergolden ratio aka Narayana's cow constant (cf. https://en.wikipedia.org/wiki/Supergolden_ratio), I noticed the following construction involving that constant and a 120-degree ($2\pi/3$ radian) triangle on that page:

120 degree triangle

I then began to wonder about similar constructions with other fractional angles: $2\pi/2$, $2\pi/4$, ...

I found that an angle of $2\pi/2$ gives a degenerate triangle based on the golden ratio:

enter image description here

$2\pi/4$ gives a right triangle based on the square-root of the golden ratio:

enter image description here

And $2\pi/6$ gives an equilateral triangle based on unity:

enter image description here

Nothing too surprising or new in the results so far, but an angle of $2\pi/5$ (72-degrees) produced the following:

enter image description here

So, the positive solution to $-2 - x + \sqrt 5 x - 2 x^2 + 2 x^4=0$, approximate value $1.11716..$, minimal polynomial $1 + x + x^2 + x^3 - x^4 - x^5 - 2 x^6 + x^8$

Is this a known/named constant?

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  • $\begingroup$ It is not a constant from this list, but this is only a small list... $\endgroup$ Commented Jan 14, 2019 at 20:58
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    $\begingroup$ The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111\ldots$. $\endgroup$
    – Blue
    Commented Jan 14, 2019 at 21:05
  • $\begingroup$ I was getting around to this question myself. Pi/4 is related to the square root of $1 - x^2 - 2 x^3 + x^4=0$. I can also relate Pi/3 to the square root of the root for $1 - x - x^2 - 2 x^3 + x^4=0$. $\endgroup$
    – Ed Pegg
    Commented Jan 14, 2019 at 22:06
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    $\begingroup$ The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1 $\endgroup$
    – Ed Pegg
    Commented Jan 14, 2019 at 22:24
  • $\begingroup$ The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org. $\endgroup$
    – Ed Pegg
    Commented Jan 14, 2019 at 23:11

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