OFFSET
-1,1
COMMENTS
This constant gives the ratio of the area between three touching circles, one with half of the radius of the two others, and the area of one of the large circular disks.
See A343235 for the same problem with three identical circular disks, where also links on the Descartes-Steiner five circle theorem and the Soddy circles are given.
The isosceles triangle with the centers of the circles as corners has two angles alpha = arctan(sqrt(5)/2) = A228496 (about 48.2 degrees).
The ratio of the perimeter of the boundary of this circular cuspodial triangle and the perimeter of the large circle is alpha/(2*Pi) + 1/4 = 0.3838602364...
FORMULA
Equals A/(Pi*r^2) = (sqrt(5)/Pi - 3*arctan(sqrt(5)/2)/(2*Pi) - 1/4)/2, where A is the area between three mutually touching circular disks with radii r, r, and r/2 (in some length unit).
Equals sqrt(5)/(2*Pi) - 3*A228496/(4*Pi) - 1/8.
EXAMPLE
0.03009091710766602117945599124597761...
MATHEMATICA
RealDigits[(Sqrt[20] - 3*ArcCos[2/3])/(4*Pi) - 1/8, 10, 100][[1]] (* Amiram Eldar, Apr 20 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 20 2021
STATUS
approved