Polygon Rings

Abstract

In general, a main position of a flexagon is a position that is, in appearance, a ring of convex polygons (Section 1.1). Consequently, an understanding of the properties of polygon rings is needed for an understanding of some of the properties of flexagons. Polygon rings are clearly defined geometric objects that exist in infinite series. In this chapter it is taken as understood that only the first few members of an infinite series are being described. Polygon rings are described as flat, slant and skew (Section 1.1). These descriptions can also be applied to main positions of flexagons, and they are described as flat main positions, slant main positions and skew main positions. All the polygon rings described in this chapter, and in other chapters, are hinged. What is meant by an even edge ring, an odd edge ring, and an even vertex ring is defined in Section 1.1. A compound edge ring and an irregular edge ring are defined in Section 1.2. Various aspects of flat edge rings of regular polygons have been discussed by several authors (Conrad and Hartline 1962; Hirst 1995; Dunlap 1997/1998; Griffiths 2001; Pook 2003). Polygon rings illustrated in this chapter have been chosen to complement points made in the text. Polygon rings in subsequent chapters complement descriptions of flexagons.