Is this (1.11716..) a known/named constant?
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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 ($2pi/3$ radian) triangle on that page:
I then began to wonder about similar constructions with other fractional angles: $2pi/2$, $2pi/4$, ...
I found that an angle of $2pi/2$ gives a degenerate triangle based on the golden ratio:
$2pi/4$ gives a right triangle based on the square-root of the golden ratio:
And $2pi/6$ gives an equilateral triangle based on unity:
Nothing too surprising or new in the results so far, but an angle of $2pi/5$ (72-degrees) produced the following:
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?
geometry constants
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
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add a comment |
$begingroup$
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 ($2pi/3$ radian) triangle on that page:
I then began to wonder about similar constructions with other fractional angles: $2pi/2$, $2pi/4$, ...
I found that an angle of $2pi/2$ gives a degenerate triangle based on the golden ratio:
$2pi/4$ gives a right triangle based on the square-root of the golden ratio:
And $2pi/6$ gives an equilateral triangle based on unity:
Nothing too surprising or new in the results so far, but an angle of $2pi/5$ (72-degrees) produced the following:
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?
geometry constants
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
$endgroup$
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It is not a constant from this list, but this is only a small list...
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– Dietrich Burde
Jan 14 at 20:58
3
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The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
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– Blue
Jan 14 at 21:05
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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 $1 - x - x^2 - 2 x^3 + x^4=0$.
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– Ed Pegg
Jan 14 at 22:06
1
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The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
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– Ed Pegg
Jan 14 at 22:24
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The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11
add a comment |
$begingroup$
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 ($2pi/3$ radian) triangle on that page:
I then began to wonder about similar constructions with other fractional angles: $2pi/2$, $2pi/4$, ...
I found that an angle of $2pi/2$ gives a degenerate triangle based on the golden ratio:
$2pi/4$ gives a right triangle based on the square-root of the golden ratio:
And $2pi/6$ gives an equilateral triangle based on unity:
Nothing too surprising or new in the results so far, but an angle of $2pi/5$ (72-degrees) produced the following:
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?
geometry constants
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
$endgroup$
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 ($2pi/3$ radian) triangle on that page:
I then began to wonder about similar constructions with other fractional angles: $2pi/2$, $2pi/4$, ...
I found that an angle of $2pi/2$ gives a degenerate triangle based on the golden ratio:
$2pi/4$ gives a right triangle based on the square-root of the golden ratio:
And $2pi/6$ gives an equilateral triangle based on unity:
Nothing too surprising or new in the results so far, but an angle of $2pi/5$ (72-degrees) produced the following:
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?
geometry constants
geometry constants
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
edited Jan 14 at 22:15
WRSomsky
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
asked Jan 14 at 20:51
WRSomskyWRSomsky
4016
4016
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It is not a constant from this list, but this is only a small list...
$endgroup$
– Dietrich Burde
Jan 14 at 20:58
3
$begingroup$
The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
$endgroup$
– Blue
Jan 14 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
Jan 14 at 22:06
1
$begingroup$
The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
$endgroup$
– Ed Pegg
Jan 14 at 22:24
$begingroup$
The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11
add a comment |
$begingroup$
It is not a constant from this list, but this is only a small list...
$endgroup$
– Dietrich Burde
Jan 14 at 20:58
3
$begingroup$
The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
$endgroup$
– Blue
Jan 14 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
Jan 14 at 22:06
1
$begingroup$
The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
$endgroup$
– Ed Pegg
Jan 14 at 22:24
$begingroup$
The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11
$begingroup$
It is not a constant from this list, but this is only a small list...
$endgroup$
– Dietrich Burde
Jan 14 at 20:58
$begingroup$
It is not a constant from this list, but this is only a small list...
$endgroup$
– Dietrich Burde
Jan 14 at 20:58
3
3
$begingroup$
The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
$endgroup$
– Blue
Jan 14 at 21:05
$begingroup$
The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
$endgroup$
– Blue
Jan 14 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
Jan 14 at 22:06
$begingroup$
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
Jan 14 at 22:06
1
1
$begingroup$
The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
$endgroup$
– Ed Pegg
Jan 14 at 22:24
$begingroup$
The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
$endgroup$
– Ed Pegg
Jan 14 at 22:24
$begingroup$
The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11
$begingroup$
The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11
add a comment |
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$begingroup$
It is not a constant from this list, but this is only a small list...
$endgroup$
– Dietrich Burde
Jan 14 at 20:58
3
$begingroup$
The Inverse Symbolic Calculator doesn't have a match for the longer decimal expansion, $1.11715586807111ldots$.
$endgroup$
– Blue
Jan 14 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
Jan 14 at 22:06
1
$begingroup$
The polynomial is known. lmfdb.org/NumberField/8.4.42903125.1
$endgroup$
– Ed Pegg
Jan 14 at 22:24
$begingroup$
The polynomials involved for Pi/7 and Pi/9 are not known at lmfdb.org.
$endgroup$
– Ed Pegg
Jan 14 at 23:11