H-triangle with k interior H-points and 3k+5 boundary H-points

Abstract

In order to study the boundary characteristics and geometric structure of an H-polygon with k interior H-points，the relationship between the interior H-points and boundary H-points of the H-triangle in a regular hexagonal Archimedean tiling was studied. Firstly, by analyzing the triple (α,β,γ) of H-triangle, the triples that may meet the requirements were determined. Then, the impossible triples were excluded by using the theory of level and the distribution characteristics of tiling points. Finally, H-triangle with k interior H-points and 3k+5 boundary H-points was obtained. Considering there were only two types of configurations, the specific constructions of these two configurations were given. The results show that the configurations of the H-triangle which satisfies the given number of boundary H-points are certain under the condition that all possible triples of triangle can be determined. The research results enrich the related theories of Archimedean tiling, and provide an important theoretical basis for the research of related problems of Archimedean tiling.

Keywords

• discrete mathematics; discrete geometry; archimedean tiling; regular hexagonal tiling; h-triangle; h-point