Elementary geometry on the integer lattice

Abstract

The n-dimensional integer lattice, denoted by $${{\mathbb {Z}}}^n$$ , is the subset of $${{\mathbb {R}}}^n$$ consisting of those points whose coordinates are all integers. In this expository paper, many concrete, intuitive, and geometric results concerning the integer lattice $${{\mathbb {Z}}}^n$$ are presented, most of them together with new elementary or streamlined proofs. Some of the presented results are new, and others are improved versions of old results.