Abstract: The hexagonal and the triangular grids are duals of each other. These two grids are the first and second ones in the family of triangular grids (we can call them as one- and two-planes triangular grids). This family comes from the mapping their points to $\mathbb Z^3$. Their symmetric properties are triangular. The next grid is the three-planes triangular grid, which looks like the mix of them. In this paper we analyze this grid and its dual from view of neighbouring conditions. After this we consider the $n$-planes grid. We prove that for $n>3$ the $n$-planes triangular grids are non-planar. We also examine the `circular' three-planes grid and the higher dimensional triangular grids.
Keywords: Discrete geometry, digital geometry, digital plane, triangular grid, hexagonal grid, neighbourhood relation, planar and non-planar graphs.
Classification (MSC2000): 52C99; 68U10
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