Free polygon enumeration and the area of an integral polygon

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral triangles and parallelograms contained inside P. c © 2000 Elsevier Science B.V. All rights reserved. MSC: 52B20; 52A38; 52B05; 52B45; 05E19 It is well known that if a convex polygon P in the Euclidean plane is triangulated using f0 vertices, f1 edges, and f2 triangles, then Euler’s formula f0 − f1 + f2 = 1 (1) holds independently of which triangulation we chose for P. When applied to integral polygons (polygons with integer point vertices), variations on Euler’s formula lead to a formula for the number of vertices contained in the interior of a polygon. For integral polygons one also obtains Pick’s theorem, a formula for the area of an integral polygon rst proved at least 100 years ago [15] and since generalized by Reeve [16,17], Macdonald [11], Ehrhart [2], Hadwiger and Wills [6], and many others [3–5]. While formula (1) holds independently of which triangulation we chose for P, the input data for each instance of (1) does depend on some initial triangulation; that is, expression of the Euler formula (1) requires the numbers f0; f1; f2 for some choice of triangulation of P. ( Research supported in part by NSF grants #DMS-9626688 and #DMS-9803571. E-mail address: [email protected] (D.A. Klain). 0012-365X/00/$ see front matter c © 2000 Elsevier Science B.V. All rights reserved. PII: S0012 -365X(99)00340 -4 110 D.A. Klain /Discrete Mathematics 218 (2000) 109–119 Fig. 1. A free edge, a free triangle, and a non-free triangle (from left to right). In the present work we consider the entire family of edges, triangles, and closed convex integral polygons Q contained inside an integral polygon P, such that each convex polygon Q is free. This means that the vertices (extreme points) of Q must also be integral points, while Q should contain no integral points that are not extreme points. See Fig. 1. These free polygons are the building blocks of any convex cell decomposition of P in which the 0-cells are precisely the family of integer points P ∩Z2. We are, in some sense, considering every triangulation of P at once with the single restriction that we use the points P ∩ Z2 as vertices. It turns out that analogues of many classical Euler-type formulas hold in this context. In forthcoming articles [7,8] the author will present a general Euler-type relation for valuations on a locally nite family of polytopes in Euclidean space of arbitrary nite dimension, working in the context of free polytope enumeration. While the general theorem requires substantial machinery to prove, the case of valuations on polygons can be dealt with using a straightforward combinatorial approach. These examples provide insight into some of the most fundamental polygon functionals from an enumerative perspective not previously considered. In the present self-contained note we consider this new perspective on Euler’s classical formula using purely combinatorial techniques. Instead of considering a xed triangulation of P, and then counting the edges, faces, etc., we will count the number of free integral polygons Q of each free type contained inside P, independently of any particular convex cell decomposition. These parameters will be found to satisfy an analogue of Euler’s classical formula (Proposition 2.2), leading in turn to a formula for the number of integer points in the interior of an integral polygon (Theorem 3.1). In analogy to Pick’s theorem (Theorem 4.1), we also derive a formula for the area of an integral polygon P via the enumeration of free integral triangles and parallelograms contained inside P (Theorem 4.2).