%I #21 Jun 21 2023 06:37:41
%S 1,1,4,4,7,1,4,2,4,2,5,5,3,3,3,1,8,6,7,8,0,8,0,4,2,2,1,1,9,3,9,6,7,7,
%T 0,0,8,9,1,5,9,0,6,9,2,0,7,8,7,9,3,1,0,7,2,0,9,9,0,5,2,1,7,4,0,6,5,6,
%U 7,4,2,9,9,0,2,4,2,1,4,1,5,0,4,3,7,6,0,8,1,6,1,0,3,0,9,1,7,0,4,5
%N Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.
%C A square pyramid with a height of h and a base of size s X s has volume V = (1/3)*s^2*h, so a square pyramid of unit volume has s = sqrt(3/h), and the slant height of each of the four triangular faces is t = sqrt(h^2 + (s/2)^2) = sqrt(h^2 + 3/(4*h)), and the total area of the four faces is A = 4*(s*t/2) = sqrt(12*h^3 + 9)/h; this area is minimized at h = (3/2)^(1/3), where it reaches A = 3^(7/6)*2^(1/3).
%C If the total surface area of all five faces including the square base is to be minimized, then the resulting height is 6^(1/3) (cf. A005486). - _Jon E. Schoenfield_, Nov 11 2018
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pyramid.html">Pyramid</a>.
%F Equals (3/2)^(1/3) = (1/2)*A010584.
%F Equals A002581/A002580. - _Michel Marcus_, Oct 23 2018
%e 1.14471424255333186780804221193967700891590692078793...
%t RealDigits[Surd[3/2, 3], 10, 120][[1]] (* _Amiram Eldar_, Jun 21 2023 *)
%o (PARI) sqrtn(3/2, 3) \\ _Michel Marcus_, Oct 23 2018
%Y Cf. A002580, A002581, A005486, A010584.
%K nonn,cons
%O 1,3
%A _Jon E. Schoenfield_, Oct 22 2018
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