This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374260 #14 Jul 22 2024 15:37:06
%S A374260 1,5,3,8,2,0,7,5,1,2,1,3,8,4,4,7,3,4,9,7,1,1,4,9,6,4,7,9,4,6,2,8,9,9,
%T A374260 4,0,9,8,7,3,9,0,7,5,8,6,9,0,8,4,4,5,0,7,3,0,8,2,6,7,5,0,8,8,8,3,4,9,
%U A374260 5,4,7,2,6,8,5,3,2,8,3,4,3,5,8,9,3,3,8
%N A374260 Decimal expansion of the Euclidean length of the shortest circuit covering all the vertices of the cube [0,1]^3.
%C A374260 It has been proved that it is not possible to join the 8 vertices of a cube with a polygonal chain that has fewer than 6 edges (see Links, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, Theorem 2.2). Thus, any circuit of 6 line segments covering all the vertices of a cube has the minimum link-length (by definition).
%C A374260 Here we consider the additional constraint of minimizing the total (Euclidean) length of the minimum-link circuit (which consists of exactly 6 line segments connected at their endpoints) that joins all the vertices of the cube [0,1] X [0,1] X [0,1].
%C A374260 Let x := (1/2)*sqrt((2/3)^(2/3)*((9+sqrt(177)))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)) + (1/2)*sqrt(4*(2/(27+3*sqrt(177)))^(1/3) - (2/3)^(2/3)*(9+sqrt(177))^(1/3) + 4*sqrt(2/((2/3)^(2/3)*(9+sqrt(177))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)))) = 1.597920933550032074764705350780465558827883608091828573735862154752648..., and then let c := 1+(x+2+sqrt(2))/(2*sqrt(2)*(x+sqrt(2))).
%C A374260 A solution to the above-stated problem is provided by the 6-link circuit (1/2, 1/2, 1+x/sqrt(2))-(c,c,0)-(-c,-c,0)-(1/2,1/2, 1+x/sqrt(2))-(-c,c,0)-(c,-c,0)-(1/2, 1/2, 1+x/sqrt(2)).
%C A374260 The total (Euclidean) length of the mentioned circuit is given by 4*((2+sqrt(2)*x)/2)*(1/x+sqrt(1+1/x^2)) = which is about 11.105251123 and this value cannot be beaten by any other 6-link circuit covering all the vertices belonging to the set {0,1} X {0,1} X {0,1}. This result follows by symmetry from the optimal polygonal chain described in the comments of A373537.
%H A374260 Matematicamente.it, <a href="https://www.matematicamente.it/forum/viewtopic.php?f=36&t=238418">Problema di minimizzazione con un triangolo rettangolo</a>.
%H A374260 Roberto Rinaldi and Marco Ripà, <a href="https://arxiv.org/abs/2212.11216">Optimal cycles enclosing all the nodes of a k-dimensional hypercube</a>, arXiv:2212.11216 [math.CO], 2022.
%F A374260 Equals 2*(2+sqrt(2)*x)*(1/x+sqrt(1+1/x^2)), where x = (1/2)*sqrt((2/3)^(2/3)*((9+sqrt(177)))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)) + (1/2)*sqrt(4*(2/(27+3*sqrt(177)))^(1/3) - (2/3)^(2/3)*(9+sqrt(177))^(1/3) + 4*sqrt(2/((2/3)^(2/3)*(9+sqrt(177))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)))) = 1.59792093355003207476470...
%e A374260 15.382075121384473497114964794628994098739075869...
%o A374260 (PARI) my(x=solve(x=1.5, 1.7, 4-8*x^2-4*x^4+x^8)); 2*(sqrt(1 + 1/x^2) + 1/x)*(2 + x*sqrt(2)) \\ _Hugo Pfoertner_, Jul 01 2024
%Y A374260 Cf. A225227, A261547, A363755, A373537.
%K A374260 nonn,cons
%O A374260 2,2
%A A374260 _Marco Ripà_, Jul 01 2024