Lebesgue Minimal Problem

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/LebesgueMinimalProblem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LebesgueMinimalProblem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Hexagon

(1/12)(7(Sqrt[3])-12)//N

A1=Sqrt[3]/2;

N[A1,10]

Pál

x=(Sqrt[3]-1)/2;
h=2s;
s=(2Sqrt[3]-3)/6;
r=1/2;
A2=2/3(3-Sqrt[3]);

N[A2,10]

Sprague

phi=ArcCos[x]-Pi/3;
hp=2r Tan[phi];
l=2r Sec[phi];
a=Sqrt[3]/3;

dA32=1/2(l-2r)(h-hp);
dA31=2r^2(Tan[phi]-phi);
A3=A2-2(dA31+dA32);

{h,hp}//N

FullSimplify[Solve[{(a-h)/h==r2/r1,r1+r2==2r},{r1,r2}]]

FullSimplify[ArcCos[(Sqrt[3]-1)/2]]

FullSimplify[1/2(l-2r)(h-hp)]

{dA31,dA32,A3}//N

FullSimplify[A3]

Expand[%84]

A3=-(109/121)-(82/(121 (Sqrt[3])))+(2/121) (Sqrt[-35139+28634 (Sqrt[3])])-(π/3)+ArcCos[(1/2) (-1+(Sqrt[3]))];

N[%84]

Summary

{A1,A2,A3}//N

Bound

N[ Pi/8+Sqrt[3]/4]