Elementary geometry on the integer lattice (Q1662428)

The integer lattice \(\mathbb{Z}^n\) is the set of all vectors in \(\mathbb{R}^n\), whose coordinates are all integers. In this expository paper, the authors present many results concerning the integer lattice \(\mathbb{Z}^n\). The results presented in the paper concern the lattice polygons, the Pick's theorem, regular and lattice simplices in \(\mathbb{R}^n\), lattice angles, lattice points in a planar region and on quadratic curves. Many results are old ones, but some of them are new. Recall that a lattice polygon is called equiangular, if its internal angles are all equal. For example, it is proved in this paper, that if \(m\in \{3,4,6,8,12\}\), then an equiangular lattice \(m\)-gon exists in \(\mathbb{R}^n\) for every \(n\geq 3\) (Theorem 4.2). This generalizes the result obtained by \textit{P. R. Scott} [Coll. Math. J. 18, No. 4, 300--306 (1987; Zbl 0995.51506)]. This paper provides a comprehensive overview of the elementary geometry on the integer lattice \(\mathbb{Z}^n\).