The rectangular spiral or the $n_1 \times n_2 \times \cdots \times n_k$ Points Problem

TL;DR

This paper generalizes Ripà's square spiral solution to higher dimensions and provides bounds for the $k$-dimensional Points Problem, offering insights into optimal configurations and their characteristics.

Contribution

It extends the square spiral solution to multi-dimensional cases and establishes a non-trivial lower bound for the problem, defining a range for optimal solutions.

Findings

Derived a non-trivial lower bound for the $k$-dimensional problem

Extended Ripà's square spiral solution to higher dimensions

Provided numerical examples illustrating the bounds

Abstract

A generalization of Rip\`a's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$ Points Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we give a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.