Irregular Cycle Flexagons

Abstract

AbstractIn a regular cycle flexagon made from regular convex polygons all the main positions of a cycle have the same appearance and the same pat structure. Among others, fundamental even edge flexagons (Section 4.2.1) and fundamental point flexagons (Section 5.3.1) are made from regular convex polygons and are regular cycle flexagons. They are solitary flexagons. In an irregular cycle flexagon made from regular convex flexagons the pat structure, but not the appearance, of main positions changes as a cycle is traversed. The irregular cycle flexagons described in this chapter can be regarded as variants of regular cycle flexagons, and are all solitary flexagons. The characteristic feature of an irregular cycle flexagon is that its associated polygon, and hence the inscribed portion of its flexagon figure, is an irregular polygon with the same vertices as a regular polygon.Three families of irregular cycle flexagons are described. The first is irregular cycle even edge flexagons. Every irregular cycle even edge flexagons has a corresponding first order fundamental even edge flexagon with the same intermediate position map. The topological invariants (Section 4.2.1) are the same. In general, nets for irregular cycle even edge flexagons are irregular, that is they are not first order fundamental edge nets (Section 3.2). Standard face numbering sequences (Section 4.1.1) are not appropriate for irregular cycle even edge flexagons. Possible irregular cycle even edge flexagons can be enumerated by counting possible irregular polygons that have the same vertices as regular polygons (Conrad and Hartline 1962). These irregular polygons are possible associated polygons for irregular cycle even edge flexagons. The one possible irregular quadrilateral, and possible irregular pentagons and irregular hexagons are shown in Figs. 7.1–7.3. All have intersecting edges. Rotations and reflections are not regarded as distinct. The arbitrary type letters shown in Figs. 7.2 and 7.3 are used in descriptions of corresponding flexagons. The numbers of possible irregular polygons, with the same vertices as regular polygons, and hence numbers of irregular cycle even edge flexagons, increase rapidly with the number of edges on polygons. There are no irregular triangles, one irregular quadrilateral, two irregular pentagons and 11 irregular hexagons.