cp's OEIS Frontend

A381756 Decimal expansion of the smallest angular distance between two vertices of the equilateral square antiprism measured along the circumscribing sphere.

%I A381756 #10 Jul 29 2025 15:36:35
%S A381756 1,3,0,6,5,2,7,1,6,1,7,1,7,4,3,7,2,7,5,5,3,4,1,6,4,6,9,0,5,9,8,6,9,4,
%T A381756 7,4,4,1,6,2,8,6,1,3,9,0,1,9,9,9,2,7,8,9,0,3,1,9,6,8,8,6,5,8,5,8,9,7,
%U A381756 4,5,3,6,9,4,0,3,0,6,5,2,9,1,1,4,4,9,1,2,9,1,0
%N A381756 Decimal expansion of the smallest angular distance between two vertices of the equilateral square antiprism measured along the circumscribing sphere.
%C A381756 The equilateral square antiprism of side number n=4, lateral edge length a, and the two bases separated vertically by h has h = a*sqrt( 1-sec^2(Pi/(2n)) ) = a/2^(1/4). The 4 vertices of the top base have Cartesian coordinates (+-a/sqrt(2),0,h/2), (0,+-a/sqrt(2),h/2); the 4 vertices at the bottom base have (+-a/2,+-a/2,-h/2). The common distance of these 8 vertices from the origin is r = a*sqrt(8+2^(3/2))/4, the radius of the circumscribing sphere. The largest dot product between pairs of the 8 vertices is sqrt(2)*a^2/8 , which is equivalent to the smallest distance measured along the surface of the sphere of radius r. Dividing this dot product through r^2 gives 2^(3/2)/(8+2^(3/2)), the cosine of the angle between closest vertices. This here is the angle measured in radians.
%H A381756 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Antiprism.html">Antiprism</a>.
%F A381756 Equals arccos(1/(2^(3/2)+1)) = arcsec(A086178).
%e A381756 1.3065271617174372755341...
%p A381756 evalf( arccos(1/(2^(3/2)+1)) ) ;
%t A381756 RealDigits[ArcCos[1/(2^(3/2)+1)],10,91][[1]] (* _Stefano Spezia_, Jul 29 2025 *)
%Y A381756 Cf. A086178.
%K A381756 nonn,cons
%O A381756 1,2
%A A381756 _R. J. Mathar_, Mar 06 2025