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%I #45 Nov 04 2024 09:17:06
%S 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,
%T 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,
%U 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
%N Decimal expansion of sqrt(3)/Pi - 1/2.
%C 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.
%C The corresponding ratio for the perimeters is 1/2.
%C A crown glass window problem.
%C The boundary of this area could be called a circular cuspodial triangle. See also the figure and discussion in the Mathematics Stack Exchange link.
%C 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.
%C From _Wolfdieter Lang_, Apr 22 2021: (Start)
%C 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...
%C 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)
%C From _M. F. Hasler_, Oct 29 2024: (Start)
%C 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).
%C 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)
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3700417/hopf-umlaufsatz-theorem">Hopf Umlaufsatz-Theorem</a>.
%H Joshua Searle and others, <a href="https://groups.google.com/g/seqfan/c/QDMpzI20qXs/m/W4ZZeNFKBgAJ">Overlapping Hyperspheres</a>, SeqFan mailing list, Oct 29, 2024.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DescartesCircleTheorem.html">Descartes Circle Theorem</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InnerSoddyCircle.html">Inner Soddy Circle</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OuterSoddyCircle.html">Outer Soddy Circle</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Crown_glass_(window)">Crown glass (window)</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Descartes'_theorem">Descartes' Theorem</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F 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.
%F Equals 1/A093602 - 1/2.
%F Equals A090551/Pi. - _Hugo Pfoertner_, Oct 29 2024
%e 0.05132889542179204951132649831296944139738648036664065279936602029103...
%t RealDigits[Sqrt[3]/Pi - 1/2, 10, 120][[1]] (* _Amiram Eldar_, Jun 20 2023 *)
%o (PARI) A343235_upto(N=99)=localprec(9+N+=1); digits(10^N*sqrt(3)\Pi)[^1] \\ _M. F. Hasler_, Oct 29 2024
%Y Cf. A090551, A093602 (Pi/sqrt(3)), A176053, A246724.
%Y Cf. A132116 (continued fraction of Pi/sqrt(3) and sqrt(3)/Pi).
%K nonn,cons,easy
%O -1,1
%A _Wolfdieter Lang_, Apr 19 2021