

A343235


Decimal expansion of sqrt(3)/Pi  1/2.


4



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

1,1


COMMENTS

This is the leftover area between three mutually touching circular discs of the same radius divided by the area of the disc of one of the circles.
The corresponding ratio for the perimeters is 1/2.
A crown glass window problem.
The boundary of this area could be called a circular cuspodial triangle. See also the figure and discussion in the Mathematics Stack Exchange link.
The ratio of the radius of the inscribed and circumscribed circle of the three kissing circles and the common radius r is r_i/r = (2*sqrt(3)  3)/3 = A246724 and r_o/r = (2*sqrt(3) + 3)/3 = A176053 = 2 + A246724. These two circles are also called inner and outer Soddy circles. See the links on the DescartesSteiner five circle theorem.
If this leftover area A(r) is normalized with the area of Pi*(r_o)^2 (outer Soddy disk) instead of Pi*r^2 (one of the three touching disks) then one obtains A(r)/(Pi*(r_o)^2) = (21/2 + 36/Pi) + (21/Pi + 6)*sqrt(3) = 0.0110557466...
The leftover area from the outer Soddy disk if all four inner circular disks (the three touching disks and the inner Soddy disk) are taken away, normalized with Pi*(r_o)^2, is 159 + 92*sqrt(3) = 0.3486742963... This is an integer in the real quadratic number field Q(sqrt(3)). (End)


LINKS



FORMULA

Equals A(r)/(Pi*r^2) = sqrt(3)/Pi  1/2 = (2*sqrt(3)  Pi)/(2*Pi), where A(r) is the area between three mutually touching circular discs of the same radius r.


EXAMPLE

0.05132889542179204951132649831296944139738648036664065279936602029103...


MATHEMATICA

RealDigits[Sqrt[3]/Pi  1/2, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)


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KEYWORD



AUTHOR



STATUS

approved



