login
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

%I #47 Jul 01 2023 16:55:38

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Brahmagupta%27s_formula">Brahmagupta's formula</a>.

%F 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.)

%e 1.653745515077771929707906238366459714566223...

%p 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

%t 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)]/4,3/5<r<9/5},r,AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]

%t 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 *)

%K nonn,cons

%O 1,2

%A _Frank M Jackson_, Sep 08 2011

%E First Mathematica program fixed by _Harvey P. Dale_, Sep 10 2011

%E Second Mathematica program added by _Harvey P. Dale_, Jan 14 2019