A generalization of the hexastix arrangement to higher dimensions
| Autor: | Haugland, Jan Kristian |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 5 pages |
| Databáze: | arXiv |
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