A374149 Decimal expansion of the minimum volume of an axis-aligned bounding box which includes the shortest minimum-link polygonal chain joining all the vertices of the cube {0,1}^3.

5, 5, 4, 5, 0, 8, 4, 9, 7, 1, 8, 7, 4, 7, 3, 7, 1, 2, 0, 5, 1, 1, 4, 6, 7, 0, 8, 5, 9, 1, 4, 0, 9, 5, 2, 9, 4, 3, 0, 0, 7, 7, 2, 9, 4, 9, 5, 1, 4, 4, 0, 7, 1, 5, 5, 3, 3, 8, 6, 2, 1, 5, 5, 6, 7, 6, 3, 1, 5, 1, 1, 5, 7, 0, 4, 7, 2, 5, 6, 1, 2, 4, 2, 6, 8, 0, 1

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

Table of n, a(n) for n=1..87.

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

Cf. A001622, A134944, A225227, A244593, A261547, A363755, A373537.

Sequence in context: A332506 A019805 A201290 * A257872 A180138 A094245

Adjacent sequences: A374146 A374147 A374148 * A374150 A374151 A374152

Keyword

nonn,cons

Author

Marco Ripà, Jun 29 2024

Status

approved