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A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.

%I A340918 #11 Feb 27 2025 07:57:22
%S A340918 1,1,5,4,4,7,9,8,3,3,4,1,9,2,7,0,7,3,7,8,3,1,9,6,1,8,4,0,4,2,3,0,2,1,
%T A340918 1,1,4,4,8,9,3,0,0,4,8,7,3,6,3,3,4,2,5,1,2,2,4,1,4,2,1,4,4,1,7,0,4,4,
%U A340918 4,8,3,8,1,4,0,4,9,7,4,9,6,7,1,4,5,0,0
%N A340918 Decimal expansion of largest angular separation (in radians) between 10 points on a unit sphere.
%C A340918 In his habilitation thesis from 1963, Ludwig Danzer provided an interval from 1.1544786 to 1.1544795 (rounded) for this value.
%D A340918 Ludwig Danzer, Endliche Punktmengen auf der 2-Sphaere mit moeglichst grossem Minimalabstand. Habilitationsschrift, Universitaet Goettingen, 1963. See link for the English translation.
%H A340918 Paolo Xausa, <a href="/A340918/b340918.txt">Table of n, a(n) for n = 1..10000</a>
%H A340918 Ludwig Danzer, <a href="https://doi.org/10.1016/0012-365X(86)90002-6">Finite point-sets on S^2 with minimum distance as large as possible</a>, Discrete Mathematics, Volume 60, June-July 1986, Pages 3-66, Table 1, page 63.
%H A340918 Hugo Pfoertner, <a href="http://www.enginemonitoring.org/sphere/pages/pack10.htm">Visualization of the optimal configuration</a>, (2001).
%H A340918 Teruhisa Sugimoto and Masaharu Tanemura, <a href="https://arxiv.org/abs/1509.01768">Exact value of Tammes problem for N=10</a>, arXiv:1509.01768 [math.MG], 12 Sep 2015.
%F A340918 atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9)) + 3))
%e A340918 1.1544798334192707378319618404230211144893004873633425122414214417...
%t A340918 First[RealDigits[ArcTan[Sqrt[4/Sqrt[3]*Cos[ArcTan[Sqrt[3*229]/9]/3] + 3]], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *)
%o A340918 (PARI) atan(sqrt((4/sqrt(3))*cos((1/3)*atan(sqrt(3*229)/9))+3))
%Y A340918 Cf. A080865, A217695, A339262.
%K A340918 nonn,cons
%O A340918 1,3
%A A340918 _Hugo Pfoertner_, Jan 30 2021