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)
From M. F. Hasler, Oct 29 2024: (Start)
Without subtraction of 1/2, the decimal expansion would just have one additional leading 5: sqrt(3)/Pi = 0.5513288954... See A132116 for the continued fraction (essentially the same for the inverse), and A093602 for the decimal expansion of the inverse, Pi/sqrt(3).
In even dimension n, the fraction of an (n-1)-sphere's surface cut off by another (n-1)-sphere with same radius and center on the first sphere, has also an expression of the form 1/3 - q*sqrt(3)/pi, where q is a rational number, depending on n: q = 0, 1/4, 3/8, 9/20, 279/560, 297/560, ... for n = 2, 4, 6, ... (End)
LINKS
Mathematics Stack Exchange, Hopf Umlaufsatz-Theorem.
Joshua Searle and others, Overlapping Hyperspheres, SeqFan mailing list, Oct 29, 2024.
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, Crown glass (window).
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.
Equals A090551/Pi. - Hugo Pfoertner, Oct 29 2024
EXAMPLE
0.05132889542179204951132649831296944139738648036664065279936602029103...
MATHEMATICA
RealDigits[Sqrt[3]/Pi - 1/2, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)
PROG
(PARI) A343235_upto(N=99)=localprec(9+N+=1); digits(10^N*sqrt(3)\Pi)[^1] \\ M. F. Hasler, Oct 29 2024
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 19 2021
STATUS
approved