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arXiv · 2304.02299

On exceptionality of dimension three in terms of lattice angles

Ken Yamamoto
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Abstract

In this study, we investigate the lattice angle, which is defined as the angle between two vectors whose components are integers. We focus on the set of angles between a fixed integer vector and other integer vectors. For non-three-dimensional lattices, we proved that this set contains all lattice angles, irrespective of the fixed vector choice. In contrast, for the three-dimensional lattice, we proved that this set of angles cannot cover all lattice angles, for any fixed vector. Thus, only the three-dimensional lattice is an exception. We further provide the condition for a given three-dimensional integer vector to intersect another integer vector at a given angle, which involves a number-theoretic property of the squared norm of the given vector and the squared tangent of the given angle.

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Ken Yamamoto. 2024-11-28. On exceptionality of dimension three in terms of lattice angles. https://doi.org/10.1142/s179304212550006x

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