Regular polygons and their application to digital curves

In this note, we discuss various kinds of 2D unit cells, or surface-unit cells, made by regular polygons (simplexes) in the plane R X R. Mathematically, the plane can be divided by simplexes or regular polygons (decomposition). If we only allow one kind of surface-unit in the plane, there are only three possible choices: regular triangle (3-regular- polygon), square (4-regular-polygon), or 6-regular-polygon. Using Euler's formula for planar graphs, we give a type of topological proofs to that a closed digital curve has at least 6 points in a 3-regular-polygon decomposition plane, has at least 8 points in a 4-regular-polygon decomposition plane, and has at least 12 points in a 6-regular-polygon decomposition plane, respectively. On the other hand, a plane can also be divided by combinations of two kinds of regular polygons. We have obtained two types of {3,6}-regular-polygon combinations, two types of {4,8}-regular-polygon combinations, and one type of {3,12}-regular-polygon combination. We also discuss the application of polygons or closed paths to digital surfaces in 3D digital spaces.