The Study About The Number Of The Boundary H-points Of H-triangleon The Regular Hexagonal Tiling

Let H be the set of vertices of a [6.6.6]-tiling, that is, a tiling of the plane by regular hexagons with unit edge. A point of H is called an H-point, a simple polygon in R2 whose corners lie in H is called an H-polygon. H can be considered as the union of two disjoint triangular lattices denoted by H+(H) such that for any two points in H+(H) there exists a translation of the plane which maps one of the two points to the other and H to H. A point of H+(H) is called an H+-point (H-point). Let A denote the set of all centers of the hexagonal tiles which determine H. A point of A is called an A-point. Clearly, H+∪H-∪A forms a triangular lattice. We will denote this triangular lattice by T= H+∪∪A. For a planar H-polygon P we denote b(P)=|H∩(?)P| and i(P)=|H∩int P|, where b(P) is the number of boundary H-points of an H-polygon P and i(P) is the number of interior H-points of P.Reay and Ding initiated the research work in 1987 about the enumeration problems of H-gons with some problems solved. Next year, they obtained a new Pick-type theorem about [6.6.6]-tiling, which laid the foundation for sloving the problem of H-polygon on [6.6.6]-tiling. In recent years Kolodziejczyk obtained a series of results about the related problems, and suggested a conjecture b(P)≤3i(P)+7 about b(P) and i(P). In 2004 Kolodziejczyk proved that for an H-triangle A with exactly one interior H-point, b(A)∈{ 3,4,5,6,7,8,10}. In 2007 Wei proved that for an H-triangle A with exactly 3 interior H-points, b(A)∈{ 3,4,5,6,7,8,9,10,11,12,13,14,16}. Moreover, Wei showed that any H-triangle A with exactly k interior H-points can have at most 3k+1 H-points on its boundary and cannot have 3k+6 H-points on its boundary, and conjecture that 6(A)∈{3,4,…,3k+4,3k+5,3k+7}.In this note, we consider the cases for k≥4. We use the notion of level of an H-triangle and the properties of triples (α,β,γ) to count the number of boundary H-points of H-triangle. We prove the conjecture is true for k= 4, that is to say, if H-triangles contain 4 interior H-points, then b(A)∈{3,4,5,6,7,8,9,10,11,12,13,14,15, 16,17,19}. The conjecture is not true for k= 5, because b(Δ) can not equal 15, that is to say, b(A)∈{3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,22}. And we prove that b(A)≠3k for any H-triangle A with exactly k interior H-points, when k≥6.