A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

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, 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, 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

Offset: 1

Author

Jon E. Schoenfield, Oct 22 2018

Keywords

• nonn
• cons

Comments

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).

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

Examples

			1.14471424255333186780804221193967700891590692078793...
		

Links

• Eric Weisstein's World of Mathematics, Pyramid.

Crossrefs

Cf. A002580, A002581, A005486, A010584.

Programs

Mathematica

RealDigits[Surd[3/2, 3], 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

PARI

sqrtn(3/2, 3) \\ Michel Marcus, Oct 23 2018

Formula

Equals (3/2)^(1/3) = (1/2)*A010584.

Equals A002581/A002580. - Michel Marcus, Oct 23 2018