A generalization of the hexastix arrangement to higher dimensions

TL;DR

This paper extends the hexastix arrangement to higher dimensions using permutohedral lattices, establishing conditions for optimal line arrangements and calculating space coverage for specific dimensions.

Contribution

It introduces a higher-dimensional generalization of hexastix based on permutohedral lattices and identifies conditions for optimal arrangements.

Findings

Possible when n is a prime power

Coverage calculated for n=4 and n=5

Alternative generalization briefly discussed

Abstract

Hexastix is an arrangement of non-overlapping infinite hexagonal prisms in four different directions that cover $\frac{3}{4}$ of space. We consider a possible generalization to $n$ dimensions, based on the permutohedral lattice $A^*_n$. The central lines of the generalized prisms are going to be oriented in $n+1$ different directions (parallel to the shortest non-zero vectors of $A^*_n$). The projection of the lines oriented in any direction along that direction to a hyperplane perpendicular to it is required to be a translation of the corresponding projection of $A^*_n$, and the minimal distance between lines oriented in any two given directions should be maximal. It is shown that this is possible if $n$ is a prime power. Also, the proportion of $n$-space that is covered is calculated for $n \in \{4, 5\}$, and an alternative generalization is briefly considered.