Fundamental Edge Flexagons

Abstract

Fundamental edge flexagons are constructed from fundamental edge nets (Sections 3.2 and 3.3). Broadly, fundamental edge flexagons are the equivalent of regular polyhedra in that they are constructed from identical regular convex polygons, and have a high degree of symmetry in their structure. They also have a high degree of symmetry in their dynamic properties. Pats are, in general, fan folded piles of leaves. Alternate pats can be single leaves. Fundamental edge flexagons are solitary flexagons which means, broadly, that they are equivalent to single polyhedra. Many other types of flexagon can be regarded as modified fundamental edge flexagons, so an understanding of their structure and dynamic properties is an essential prerequisite to the understanding of flexagons in general.

Two families of fundamental edge flexagons are discussed in this chapter. One family is first order fundamental even edge flexagons. In a main position, a first order fundamental even edge flexagon is, in appearance, a regular even edge ring of 2n regular convex polygons (Section 2.2.2), where n ≥ 2. A main position can be divided into S identical sectors containing two adjacent pats, hence S = n. The curvature is an important property of main positions, and is calculated in the same way as the curvature of the corresponding ring (Section 1.1). Nets for first order fundamental even edge flexagons are derived by fan folding an appropriate first order fundamental edge net (Section 3.2) onto a regular even edge ring and applying an appropriate face numbering sequence.