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, kIP+1/2BP-1, where BP is the number of lattice points on the boundary of P in Z3, IP 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.
1. Introduction
In the Euclidean plane R2, a lattice point is one whose coordinates are both integers. A lattice polygon is a polygon with all vertices on integer coordinates. The area A(P) of a simple lattice polygon P can be given by celebrated Pick’s theorem [1]
(1)A(P)=I(P)+12B(P)-1,
where B(P) is the number of lattice points on the boundary of P and I(P) is the number of lattice points in the interior of P.
Pick’s formula can be used to compute the area of a lattice polygon conveniently.
For example, in Figure 1, I(P)=60, B(P)=15. Then, the area of the polygon is A(P)=60+15-1=74.
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
(2)0,0,0T,1,0,0T,0,1,0T,1,1,rT,
where r 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 R3. For any (a,b,c)T∈Z3 with (a,b,c)=1, that is, the greatest common factor of a,b,c is one, denote by K, ax+by+cz=0, the two-dimensional subspace of R3. Then we established the following theorem.
Theorem 1.
If P is simple lattice polygon in the K, then the area of P is
(3)k(I(P)+12B(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 the constant (a3+ab2)a2+b2+c2.
Remark 2.
Although the simple lattice polygon P is in the two-dimensional subspace K, the lattice points in P belong to Z3.
Let (a,b,c)T=(1,0,0)T in the Theorem; then we can get Pick’s theorem in some coordinate plane of R3.
Corollary 3.
If P is simple lattice polygon in the K, whose normal vector is (1,0,0)T, then the area of P is
(4)I(P)+12B(P)-1.
2. Proof of Main Result
For any (a,b,c)T∈Z3 with (a,b,c)=1, there is two-dimensional subspace of R3(5)ax+by+cz=0,
whose normal vector is just (a,b,c)T. We denote this two-dimensional subspace by K.
By the theory of linear equations system, (-b,a,0)T and (-c,0,a)T are two linearly independent solutions of (5). We denote (-b,a,0)T by α and (-c,0,a)T by β. Obviously, α and β are also the basis of K.
Lemma 4.
For any (a,b,c)T∈Z3 with (a,b,c)=1, there exists the lattice basis with the minimal area in the two-dimensional subspace K.
Proof.
The area of parallelogram generated by α and β is
(6)1a2+b2+c2a-b-cba0c0a=a3+ab2+ac2a2+b2+c2=aa2+b2+c2.
Denote (a,b,c)T by n. For any lattice basis in K, k1α+k2β and l1α+l2β, where ki,li∈Z (i=1,2) and k1l1k2l2=0. The area of parallelogram generated by k1α+k2β and l1α+l2β is
(7)1a2+b2+c2n⋮k1α+k2β⋮l1α+l2β=1a2+b2+c2(n,α,β)×1000k1l10k2l2,
where n⋮k1α+k2β⋮l1α+l2β denote the determinant of n,k1α+k2β, and l1α+l2β.
Thus the lattice basis k1α+k2β and l1α+l2β have the minimal area if and only if k1l1k2l2=1.
Let k1=1,k2=0,l1=0, and l2=1, and α,β are the lattice basis with the minimal area in the two-dimensional subspace K.
Lemma 5.
For any (a,b,c)T∈Z3 with (a,b,c)=1, there exists the orthogonal lattice basis in the two-dimensional subspace K.
Proof.
By Lemma 4, α,β are the lattice basis with the minimal area in the two-dimensional subspace K. By Schmidt orthogonalization, let
(8)γ1=α=-b,a,0T,γ2=β-(β,γ1)(γ1,γ1)=-c,0,aT-bca2+b2γ1=-c,0,aT--b2ca2+b2,abca2+b2,0T=-a2ca2+b2,-abca2+b2,a3+ab2a2+b2T,
where (β,γ1) denote the usual inner product of β,γ1 in R3.
Thus
(9)η1=γ1=α=-b,a,0T,η2=(a2+b2)γ2=-a2c,-abc,a3+ab2T
are the orthogonal lattice basis in the two-dimensional subspace K.
Proof of Theorem.
By Lemma 5, η1,η2 are the orthogonal lattice basis in the two-dimensional subspace K.
The area of parallelogram generated by η1, η2 is
(10)1a2+b2+c2a-b-a2cba-abcc0a3+ab2=a5+a3b2+ab2c2+a3c2+a3b2+ab4a2+b2+c2=a(a4+2a2b2+b4+(a2+b2)c2)a2+b2+c2=a(a2+b2)(a2+b2+c2)a2+b2+c2=(a3+ab2)a2+b2+c2,
which just is the constant k 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.
PickG. A.Geometrisches zur Zahlentheorie, Sitzungber189919311319DingR.KoŁodziejczykK.ReayJ.A new Pick-type theorem on the hexagonal lattice1988682-3171177MR9261212-s2.0-001104727910.1016/0012-365x(88)90110-0FunkenbuschW. W.From Euler’s formula to Pick’s formula using an edge theorem197481664764810.2307/2319224MR1537447GrunbaumB.ShephardG. C.Pick's theorem19931002150161MR121240110.2307/2323771LiuA. C. F.Lattice points and Pick's theorem197952423223510.2307/2689416MR545564