OFFSET
1,1
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 are considering the additional constraint that asks to minimize the volume of the Axis-Aligned Bounding Box (AABB) including the above-mentioned optimal polygonal chain consisting of only 6 connected line segments and that joins all the vertices of the cube [0,1] X [0,1] X [0,1].
Given phi = (1+sqrt(5))/2, the well-known golden ratio (see A001622), a valid polygonal chain is (0, 1, 0)-(0, 0, 0)-((1+phi)/2, 0, (1+phi)/2)-(1/2, 1+phi, 1/2)-((1-phi)/2, 0, (1+phi)/2)-(1, 0, 0)-(1, 1, 0) (see Links, p. 164), and consequently the minimum volume AABB is [(1-phi)/2, (1+phi)/2] X [0, 1+phi] X [0, (1+phi)/2].
As noted by Hugo Pfoertner, the present sequence is also given by phi^5/2 (i.e., A244593/2).
LINKS
Marco Ripà, General uncrossing covering paths inside the Axis-Aligned Bounding Box, Journal of Fundamental Mathematics and Applications, Volume 4, 2021, Number 2, Pages 154-166.
FORMULA
Equals phi*(1+phi)*((1+phi)/2), where phi := (1+sqrt(5))/2 is the golden ratio.
Equals (11+5*sqrt(5))/4.
Equals phi^5/2.
Equals 10*A134944 + 3/2.
EXAMPLE
5.5450849718747371205114670859140952943...
MATHEMATICA
RealDigits[(11+5*Sqrt[5])/4, 10, 100][[1]].
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Marco Ripà, Jun 29 2024
STATUS
approved