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A256099
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Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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1.5436890126920763615708559...
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MATHEMATICA
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PROG
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(PARI) solve(x=1, 2, x^3-2*x^2+2*x-2) \\ Michel Marcus, Oct 24 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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