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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 (list; constant; graph; refs; listen; history; text; internal format)
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 Descartes-Steiner five circle theorem.
From Wolfdieter Lang, Apr 22 2021: (Start)
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
Mathematics Stack Exchange, Hopf Umlaufsatz-Theorem.
Eric Weisstein's World of Mathematics, Descartes Circle Theorem.
Eric Weisstein's World of Mathematics, Inner Soddy Circle.
Eric Weisstein's World of Mathematics, Outer Soddy Circle.
Wikipedia, Descartes' Theorem.
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.
Equals 1/A093602 - 1/2.
EXAMPLE
0.05132889542179204951132649831296944139738648036664065279936602029103...
MATHEMATICA
RealDigits[Sqrt[3]/Pi - 1/2, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)
CROSSREFS
Sequence in context: A082343 A166125 A199074 * A153457 A057778 A071545
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Apr 19 2021
STATUS
approved

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Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)