Pick's Theorem in Two-Dimensional Subspace of ℝ3

In the Euclidean space ℝ3, denote the set of all points with integer coordinate by ℤ3. For any two-dimensional simple lattice polygon P, we establish the following analogy version of Pick's Theorem, k(I(P) + (1/2)B(P) − 1), where B(P) is the number of lattice points on the boundary of P in ℤ3, I(P) is the number of lattice points in the interior of P in ℤ3, and k is a constant only related to the two-dimensional subspace including P.


Introduction
In the Euclidean plane R 2 , 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] ( ) = ( ) + 1 2 ( ) − 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.
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 twodimensional subspace of R 3 . For any ( , , ) ∈ Z 3 with ( , , ) = 1, that is, the greatest common factor of , , is one, denote by , + + = 0, the two-dimensional subspace of R 3 . 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 Z 3 , ( ) is the number of lattice points in the interior of in Z 3 , and is the constant Remark 2. Although the simple lattice polygon is in the two-dimensional subspace , the lattice points in belong to Z 3 .

Corollary 3.
If is simple lattice polygon in the , whose normal vector is (1, 0, 0) , then the area of is
Proof. By Lemma 4, , are the lattice basis with the minimal area in the two-dimensional subspace . By Schmidt orthogonalization, let where ( , 1 ) denote the usual inner product of , 1 in R 3 . Thus are the orthogonal lattice basis in the two-dimensional subspace .
Proof of Theorem. By Lemma 5, 1 , 2 are the orthogonal lattice basis in the two-dimensional subspace . The area of parallelogram generated by 1 , 2 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.