Abstract
In the Euclidean space , denote the set of all points with integer coordinate by . For any two-dimensional simple lattice polygon , we establish the following analogy version of Pick’s Theorem, , where is the number of lattice points on the boundary of in , is the number of lattice points in the interior of in , and is a constant only related to the two-dimensional subspace including .
1. Introduction
In the Euclidean plane , a lattice point is one whose coordinates are both integers. A lattice polygon is a polygon with all vertices on integer coordinates. The area of a simple lattice polygon can be given by celebrated Pick’s theorem [1] where is the number of lattice points on the boundary of and is the number of lattice points in the interior of .
Pick’s formula can be used to compute the area of a lattice polygon conveniently.
For example, in Figure 1, , . Then, the area of the polygon is .
There are many papers concerning Pick’s theorem and its generalizations [2–5], which mostly be discussed in two dimensions.
Unfortunately, Pick theorem is failed in three dimensions. In 1957, John Reeve found a class of tetrahedra, named as Reeve tetrahedra later, whose vertices are where is a positive integer.
All Reeve tetrahedra contain the same number of lattice points, but their volumes are different.
In this note, we discussed Pick’s theorem in two-dimensional subspace of . For any with , that is, the greatest common factor of is one, denote by , , the two-dimensional subspace of . Then we established the following theorem.
Theorem 1. If is simple lattice polygon in the , then the area of is where is the number of lattice points on the boundary of in , is the number of lattice points in the interior of in , and is the constant .
Remark 2. Although the simple lattice polygon is in the two-dimensional subspace , the lattice points in belong to .
Let in the Theorem; then we can get Pick’s theorem in some coordinate plane of .
Corollary 3. If is simple lattice polygon in the , whose normal vector is , then the area of is
2. Proof of Main Result
For any with , there is two-dimensional subspace of whose normal vector is just . We denote this two-dimensional subspace by .
By the theory of linear equations system, and are two linearly independent solutions of (5). We denote by and by . Obviously, and are also the basis of .
Lemma 4. For any with , there exists the lattice basis with the minimal area in the two-dimensional subspace .
Proof. The area of parallelogram generated by and is
Denote by . For any lattice basis in , and , where () and . The area of parallelogram generated by and is
where denote the determinant of , and .
Thus the lattice basis and have the minimal area if and only if .
Let , and , and are the lattice basis with the minimal area in the two-dimensional subspace .
Lemma 5. For any with , there exists the orthogonal lattice basis in the two-dimensional subspace .
Proof. By Lemma 4, are the lattice basis with the minimal area in the two-dimensional subspace . By Schmidt orthogonalization, let
where denote the usual inner product of in .
Thus
are the orthogonal lattice basis in the two-dimensional subspace .
Proof of Theorem. By Lemma 5, are the orthogonal lattice basis in the two-dimensional subspace .
The area of parallelogram generated by , is
which just is the constant in the theorem.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Beijing Higher Education Young Elite Teacher Project (Grant no. YETP0770), the National Natural Science Foundation of China (Grant no. 11001014), and the Young Teachers Domestic Visiting Scholars Program of Beijing Forestry University.