A generalization of the hexastix arrangement to higher dimensions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917748216430592 |
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| author | Haugland, Jan Kristian |
| author_facet | Haugland, Jan Kristian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of the hexastix arrangement to higher dimensions Haugland, Jan Kristian Combinatorics Metric Geometry 05B40 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. |
| title | A generalization of the hexastix arrangement to higher dimensions |
| topic | Combinatorics Metric Geometry 05B40 |
| url | https://arxiv.org/abs/2408.07112 |