A374260 Decimal expansion of the Euclidean length of the shortest circuit covering all the vertices of the cube [0,1]^3.

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, 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, 5, 4, 7, 2, 6, 8, 5, 3, 2, 8, 3, 4, 3, 5, 8, 9, 3, 3, 8

Offset: 2

Author

Marco Ripà, Jul 01 2024

Keywords

• nonn
• cons

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). Thus, any circuit of 6 line segments covering all the vertices of a cube has the minimum link-length (by definition).

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].

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))).

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)).

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.

Examples

			15.382075121384473497114964794628994098739075869...
		

Links

• Matematicamente.it, Problema di minimizzazione con un triangolo rettangolo.
• Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.

Crossrefs

Cf. A225227, A261547, A363755, A373537.

Programs

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

Formula

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...