Fundamental Compound Edge Flexagons

Abstract

Compound edge flexagons are even edge flexagons with main positions that are, in appearance, compound edge rings of regular convex polygons (Section 2.2.4). In a compound edge ring, alternate hinge angles (Fig. 1.3) are the same and, in general, alternate polygons are the same distance from the centre of the ring. Compound edge flexagons with at least some flat main positions were briefly described by Conrad and Hartline (1962) and in more detail by Pook (2003). A compound edge ring of 2n regular convex polygons, and corresponding compound edge flexagons, can be divided into n identical sectors each containing two polygons. All the possible flat non overlapping, compound edge rings, containing up to ten polygons, with up to 12 edges on the constituent polygons are listed in Table 2.5. Possible non flat compound edge rings of four regular polygons, with up to eight edges on the constituent polygons, are listed in Table 2.6.

Fundamental compound edge flexagons are made from first order fundamental edge nets (Section 3.2), and a standard face numbering sequence is used (Section 4.1.1). They are regular cycle flexagons in which all the main positions of a cycle have the same appearance and the same pat structure. They have some analogies with first order fundamental even edge flexagons (Section 4.2.1). In particular, they are solitary flexagons and the topological invariants are the same. Leaves usually have to be bent during flexing, although there are exceptions. Most fundamental compound edge flexagons are too difficult to flex to be aesthetically satisfying, and are therefore of only theoretical interest. Various flexes are used but there is no characteristic flex that can be used for all fundamental compound edge flexagons.