OFFSET
1,2
COMMENTS
This geometrical problem is considered in the Alten et al. reference on pp. 190-192.
The geometrical problem is to find in the first quadrant the point P on a circle (radius R) such that the ratio of the normal to the y-axis through P and the radius equals the ratio of the segments of the radius on the y-axis. See the link with a figure and more details. For Omar Khayyám see the references as well as the Wikipedia and MacTutor Archive links.
The ratio of the length of the normal x and the segment h on the y-axis starting at the origin is called xtilde, and satisfies the cubic equation
xtilde^3 -2*xtilde^2 + 2*xtilde - 2 = 0. This xtilde is the tangent of the angle alpha between the positive y-axis and the radius vector from the origin to the point P. This cubic equation has only one real solution xtilde = tan(alpha) given in the formula section. The present decimal expansion belongs to xtilde.
Apart from the first digit the same as A192918. - R. J. Mathar, Apr 14 2015
REFERENCES
H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, pp. 190-192.
O. Khayyam, A paper of Omar Khayyam, Scripta Math. 26 (1963), 323-337.
LINKS
Wolfdieter Lang, A Geometrical Problem of Omar Khayyám and its Cubic.
MacTutor History of Mathematics archive, Omar Khayyám
Wikipedia, Omar Khayyám
FORMULA
xtilde = tan(alpha) = ((3*sqrt(33) + 17)^(1/3) - (3*sqrt(33) - 17)^(1/3) + 2)/3 = 1.54368901269...
The corresponding angle alpha is approximately 57.065 degrees.
The real root of x^3-2*x^2+2*x-2. Equals tau^2-tau where tau is the tribonacci constant A058265. - N. J. A. Sloane, Jun 19 2019
EXAMPLE
1.5436890126920763615708559...
MATHEMATICA
RealDigits[Root[x^3 - 2 x^2 + 2 x - 2, 1], 10, 105][[1]] (* Jean-François Alcover, Oct 24 2019 *)
PROG
(PARI) solve(x=1, 2, x^3-2*x^2+2*x-2) \\ Michel Marcus, Oct 24 2019
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 08 2015
STATUS
approved