Summary: The two-dimensional hexagonal grid and the three-dimensional face-centered cubic grid can be described by intersecting $\Bbb Z ^{3}$ and $\Bbb Z ^{4}$ with (hyper)planes. Corresponding grids in higher dimensions are examined. Also, we explain the connection between a number of well-known three-dimensional grids by using this construction. The union of four hyperplanes (in a circular way) gives the bcc grid. Based on these connections, several types of neighborhood structures are introduced on these grids. These structures span from the most natural ones (crystal bonds, Voronoi neighbors) to infinite families. In this paper, we define path-based distance functions on the high-dimensional generalizations of the hexagonal grid.