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 A373537 #26 Jul 02 2024 05:02:57
%S A373537 1,1,1,0,5,2,5,1,1,2,3,0,6,5,3,3,1,7,9,7,3,5,9,1,7,1,1,2,1,5,2,4,1,9,
%T A373537 5,1,2,7,9,3,9,2,0,9,8,0,9,9,1,9,1,7,3,4,3,8,5,9,0,0,5,5,1,8,2,1,6,5,
%U A373537 5,0,6,1,1,2,7,2,8,5,2,4,2,1,8,3,1,7,3
%N A373537 Decimal expansion of the Euclidean length of the shortest minimum-link polygonal chains joining all the vertices of the cube [0,1]^3.
%C A373537 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).
%C A373537 Here we consider the additional constraint of minimizing the total (Euclidean) length of the minimum-link polygonal chains (which consist of exactly 6 line segments connected at their endpoints) that join all the vertices of the cube [0,1] X [0,1] X [0,1].
%C A373537 A solution to the above-stated problem is provided by the 6-link polygonal chain (0,0,1)-(0,0,0)-(1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),0)-(1/2,1/2,1+x/sqrt(2))-(- (x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),0)-(1,0,0)-(1,0,1), 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)))) = 1.597920933550032074764705350780465558827883608091828573735862154752648...
%C A373537 The total (Euclidean) length of the mentioned polygonal chain is about 11.105251123 and this value cannot be beaten by any other 6-link polygonal chain covering all the vertices belonging to the set {0,1} X {0,1} X {0,1} (a nice proof was posted on MathOverflow on June 5, 2024 by a new user, DR.LL, whose profile was subsequently deleted for unknown reasons).
%H A373537 Matematicamente.it, <a href="https://www.matematicamente.it/forum/viewtopic.php?f=36&t=238418">Problema di minimizzazione con un triangolo rettangoloe</a>.
%H A373537 Math Overflow, <a href="https://mathoverflow.net/questions/472583/shortest-polygonal-chain-with-6-edges-visiting-all-the-vertices-of-a-cube">Shortest polygonal chain with 6 edges visiting all the vertices of a cube</a>.
%H A373537 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 A373537 Equals 2*(1+1/sqrt(2)+((2+sqrt(2)*x)/2)*(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 A373537 11.10525112306533179735917112152419512793920980991917343859...
%o A373537 (PARI) my(x=solve(x=1.5,1.7,4-8*x^2-4*x^4+x^8)); 2 + sqrt(2) + (sqrt(1 + 1/x^2) + 1/x) * (2 + sqrt(2)*x) \\ _Hugo Pfoertner_, Jun 10 2024
%Y A373537 Cf. A225227, A261547, A363755.
%K A373537 nonn,cons
%O A373537 2,5
%A A373537 _Marco Ripà_, Jun 08 2024