OFFSET
2,5
COMMENTS
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).
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].
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...
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).
LINKS
Matematicamente.it, Problema di minimizzazione con un triangolo rettangoloe.
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
FORMULA
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...
EXAMPLE
11.10525112306533179735917112152419512793920980991917343859...
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Marco Ripà, Jun 08 2024
STATUS
approved