A193211 Decimal expansion of the value of r that maximizes the Brahmagupta expression sqrt((-1+r+r^2+r^3)*(1-r+r^2+r^3)*(1+r-r^2+r^3)*(1+r+r^2-r^3))/4.

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

Offset: 1

Author

Frank M Jackson, Sep 08 2011

Keywords

• nonn
• cons

Comments

The area of a convex quadrilateral with fixed sides is maximal when it is organized as a convex cyclic quadrilateral. Furthermore in order that a quadrilateral can have sides in a geometric progression 1:r:r^2:r^3 its common ratio r is limited to the range 1/t < r < t where t is the tribonacci constant (A058265). Consequently when r=1.6537455... it maximizes Brahmagupta's expression for the area of a convex cyclic quadrilateral whose sides form a geometric progression.

Examples

			1.653745515077771929707906238366459714566223...
		

Links

• Wikipedia, Brahmagupta's formula.

Programs

Maple

Digits:=200; fsolve( -3*r^10+5*r^8+2*r^6+18*r^4+r^2+1, r ); # N. J. A. Sloane, Jan 14 2019

Mathematica

RealDigits[r/.NMaximize[{Sqrt[(-1+r+r^2+r^3)(1-r+r^2+r^3)(1+r-r^2+r^3)(1+r+r^2-r^3)]/4,3/5120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]
RealDigits[r/.FindRoot[1+r^2+18r^4+2r^6+5r^8-3r^10==0,{r,2}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Jan 14 2019 *)

Formula

r is the positive real root of the equation 1 + r^2 + 18*r^4 + 2*r^6 + 5*r^8 - 3*r^10 = 0. (Corrected by N. J. A. Sloane, Jan 14 2019. Thanks to Harvey P. Dale for pointing that the old expression was incorrect.)

Extensions

First Mathematica program fixed by Harvey P. Dale, Sep 10 2011

Second Mathematica program added by Harvey P. Dale, Jan 14 2019