A270230 - OEIS

%I #22 Oct 01 2022 15:35:13

%S 2,3,8,7,3,2,4,1,4,6,3,7,8,4,3,0,0,3,6,5,3,3,2,5,6,4,5,0,5,8,7,7,1,5,

%T 4,3,0,5,1,6,8,9,4,6,8,6,1,0,6,8,4,6,7,3,1,2,1,5,0,1,0,1,6,0,8,8,3,4,

%U 5,1,9,6,4,5,1,3,3,9,8,0,2,6,3,5,1,7,0,7,0,4,1,4,9,3,7,9,6,2,8,9,3,4,1,0,9

%N Decimal expansion of 3/(4*Pi).

%C Consider generic prisms with triangular bases (tp), enclosed by a sphere, and let f(tp) be the fraction of the sphere volume occupied by any of them (i.e., the ratio of the prism volume to the sphere volume). Then this constant is the supremum of f(tp). It is attained by prisms which have as their base equilateral triangles with edge lengths r*sqrt(2), and rectangular side faces that are r*sqrt(2) wide and r*2/sqrt(3) high, where r is the radius of the enclosing, circumscribed sphere.

%C An intriguing fact is that the volume of such a best-fitting prism is exactly r^3. Hence, 1/a is the volume of a sphere with radius 1.

%C Examples of similar constants obtained for other shapes enclosed by spheres are: A020760 for cylinders and A165952 for cuboids.

%H Stanislav Sykora, <a href="/A270230/b270230.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.238732414637843003653325645058771543051689468610684673121501016...

%t First@ RealDigits[N[3/4/Pi, 120]] (* _Michael De Vlieger_, Mar 15 2016 *)

%o (PARI) 3/4/Pi

%Y Cf. A002193, A019699 (one tenth of 1/a), A020760, A020832 (one tenth of 2/sqrt(3)), A165952.

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, Mar 13 2016