Integer geometry: some examples and constructions

Abstract

Integer Geometry is the geometry of points that are pairwise integer distances apart. An integer polygon is a convex set of n points in the plane such that no three points are collinear and the distance between any two points is an integer. Similarly an integer polyhedron is a convex set of n points in space such that no three are collinear, each face is an integer polygon, no two faces are coplaner, and the points are all integer distances apart. The measure used to order such polygons and polyhedra is perimeter-plus which is the sum of all of the edges added to the sum of all of the diagonals [1]. The purpose of this paper is to provide pictures of some fundamental examples as well as a brief explanation of some of the constructions. We begin by looking at polygons and then use those to build polyhedra and conclude by examining two closely related examples of sixfaced polyhedra.