On exceptionality of dimension three in terms of lattice angles

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.