A256099 Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem.

1, 5, 4, 3, 6, 8, 9, 0, 1, 2, 6, 9, 2, 0, 7, 6, 3, 6, 1, 5, 7, 0, 8, 5, 5, 9, 7, 1, 8, 0, 1, 7, 4, 7, 9, 8, 6, 5, 2, 5, 2, 0, 3, 2, 9, 7, 6, 5, 0, 9, 8, 3, 9, 3, 5, 2, 4, 0, 8, 0, 4, 0, 3, 7, 8, 3, 1, 1, 6, 8, 6, 7, 3, 9, 2, 7, 9, 7, 3, 8, 6, 6, 4, 8, 5, 1, 5, 7, 9, 1, 4, 5, 7, 6, 0, 5, 9, 1

Offset: 1

Author

Wolfdieter Lang, Apr 08 2015

Keywords

• nonn
• cons
• easy

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

Examples

			1.5436890126920763615708559...
		

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

Crossrefs

Cf. A058265. Essentially the same as A192918.

Programs

Mathematica

RealDigits[Root[x^3 - 2 x^2 + 2 x - 2, 1], 10, 105][[1]] (* Jean-François Alcover, Oct 24 2019 *)

PARI

solve(x=1, 2, x^3-2*x^2+2*x-2) \\ Michel Marcus, Oct 24 2019

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