cp's OEIS Frontend

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

%I A256099 #34 Feb 16 2023 09:47:04
%S A256099 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,
%T A256099 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,
%U A256099 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
%N A256099 Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem.
%C A256099 This geometrical problem is considered in the Alten et al. reference on pp. 190-192.
%C A256099 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.
%C A256099 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
%C A256099   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.
%C A256099 Apart from the first digit the same as A192918. - _R. J. Mathar_, Apr 14 2015
%D A256099 H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, pp. 190-192.
%D A256099 O. Khayyam, A paper of Omar Khayyam, Scripta Math. 26 (1963), 323-337.
%H A256099 Wolfdieter Lang, <a href="/A256099/a256099_1.pdf">A Geometrical Problem of Omar Khayyám and its Cubic</a>.
%H A256099 MacTutor History of Mathematics archive, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Khayyam.html"> Omar Khayyám</a>
%H A256099 Wikipedia, <a href="https://en.wikipedia.org/wiki/Omar_Khayy%C3%A1m"> Omar Khayyám</a>
%F A256099 xtilde = tan(alpha) = ((3*sqrt(33) + 17)^(1/3) - (3*sqrt(33) - 17)^(1/3) + 2)/3 = 1.54368901269...
%F A256099 The corresponding angle alpha is approximately 57.065 degrees.
%F A256099 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
%e A256099 1.5436890126920763615708559...
%t A256099 RealDigits[Root[x^3 - 2 x^2 + 2 x - 2, 1], 10, 105][[1]] (* _Jean-François Alcover_, Oct 24 2019 *)
%o A256099 (PARI) solve(x=1, 2, x^3-2*x^2+2*x-2) \\ _Michel Marcus_, Oct 24 2019
%Y A256099 Cf. A058265. Essentially the same as A192918.
%K A256099 nonn,cons,easy
%O A256099 1,2
%A A256099 _Wolfdieter Lang_, Apr 08 2015