User:Etewilak/sandbox

Sunday, 8 August 2241 ראשון ל׳ באב ו׳״א May a total solar eclipse be visible from Jerusalem on the date mentioned above.

See https://eclipse.gsfc.nasa.gov/SEsearch/SEsearchmap.php?Ecl=22410808 for the map of the solar eclipse of 8 August 2241.

Hebrew calendar
I’ve been longing for the arrival of a 384-day Hebrew year The Hebrew year 5782 will be the first 384-day Hebrew year since 5755.

Formulas
The real root of the cubic equation $$ x^3+2x^2+10x-20=0 $$ is given as follows:

$$ x = \dfrac{\sqrt[3]{352+6\sqrt{3930}}+\sqrt[3]{352-6\sqrt{3930}}-2}{6} \approx 1.368808108 $$

It can also be expressed in terms of hyperbolic sine and its inverse:

$$ x = \dfrac{-2+2\sqrt{26}\sinh{\dfrac{\sinh^{-1}{\dfrac{88\sqrt{26}}{169}}}{3}}}{3} $$

Plastic number
The plastic number, denoted by $$\rho$$, is the real root of the cubic equation $$ x^3-x-1=0 $$.

$$ \rho = \dfrac{\sqrt[3]{108+12\sqrt{69}}+\sqrt[3]{108-12\sqrt{69}}}{6} \approx 1.324717957 $$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$$ \rho = \dfrac{2\sqrt{3}\cosh{\dfrac{\cosh^{-1}{\dfrac{3\sqrt{3}}{2}}}{3}}}{3} $$

Its algebraic conjuagates are $$ A \pm Bi $$, where

$$ A = \frac{-\sqrt[3]{108+12\sqrt{69}}-\sqrt[3]{108-12\sqrt{69}}}{12} \approx -0.662358979 $$

$$ B = \frac{\sqrt[3]{12\sqrt{3}+4\sqrt{23}}-\sqrt[3]{12\sqrt{3}-4\sqrt{23}}}{4} \approx 0.562279512 $$

Each complex conjugate has an absolute value of

$$ \frac{\sqrt{6\sqrt[3]{100-12\sqrt{69}}+6\sqrt[3]{100+12\sqrt{69}}-12}}{6} \approx 0.868836962 $$

Supergolden ratio
The supergolden ratio, denoted by $$\psi$$, is the real root of the cubic equation $$x^3-x^2-1=0$$.

$$ \psi = \frac{2+\sqrt[3]{116+12\sqrt{93}}+\sqrt[3]{116-12\sqrt{93}}}{6} \approx 1.465571232 $$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$$ \psi = \dfrac{1+2\cosh{\dfrac{\cosh^{-1}{\dfrac{29}{2}}}{3}}}{3} $$

Its algebraic conjugates are $$A \pm Bi$$, where

$$ A = \frac{4-\sqrt[3]{116+12\sqrt{93}}-\sqrt[3]{116-12\sqrt{93}}}{12} \approx -0.232785616 $$

$$ B = \frac{\sqrt[3]{348\sqrt{3}+108\sqrt{31}}-\sqrt[3]{348\sqrt{3}-108\sqrt{31}}}{12} \approx 0.792551993 $$

Each complex conjugate has an absolute value of

$$ \frac{\sqrt{6\sqrt[3]{108+12\sqrt{93}}+6\sqrt[3]{108-12\sqrt{93}}}}{6} \approx 0.826031358 $$

Tribonacci constant
The tribonacci constant is the real root of the cubic equation $$x^3-x^2-x-1=0$$.

$$ x = \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3} \approx 1.839286755 $$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$$ x = \dfrac{1+4\cosh{\dfrac{\cosh^{-1}{\dfrac{19}{8}}}{3}}}{3} $$

Its algebraic conjugates are $$ A \pm Bi $$, where

$$ A = \frac{2-\sqrt[3]{19+3\sqrt{33}}-\sqrt[3]{19-3\sqrt{33}}}{6} \approx -0.419643378 $$

$$ B = \frac{\sqrt[3]{57\sqrt{3}+27\sqrt{11}}-\sqrt[3]{57\sqrt{3}-27\sqrt{11}}}{6} \approx 0.606290729 $$

Each complex conjugate has an absolute value of

$$ \frac{\sqrt{3\sqrt[3]{17+3\sqrt{33}}+3\sqrt[3]{17-3\sqrt{33}}-3}}{3} \approx 0.737352706 $$