Irregular Ring Even Edge Flexagons

Abstract

AbstractIrregular ring even edge flexagons are even edge flexagons with main positions that are, in appearance, irregular even edge rings of regular convex polygons (Section 2.2.5). An even edge ring consists of an even number of edge hinged polygons (Section 1.1). An irregular even edge ring is an even edge ring that is neither regular (Section 2.2.2) nor compound (Section 2.2.4). Identical sectors in irregular even edge rings contain at least three polygons. For example, the flat irregular even edge ring of six regular pentagons (Fig. 2.17) has two sectors with three pentagons in each. The flat irregular ring of eight squares type A (Fig. 9.1a) has two sectors with four squares in each, but type B (Fig. 9.1b) cannot be divided so it has one sector containing eight squares. Descriptions in this chapter are largely restricted to flat non overlapping irregular even edge rings and corresponding flexagons.Fundamental irregular ring even 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. Fundamental irregular ring even edge flexagons can be regarded as generalisations of fundamental compound edge flexagons, so they are solitary flexagons, and the topological invariants are the same (Sections 4.2.1 and 6.2). Various flexes are used but there is no characteristic flex that can be used for all irregular ring even edge flexagons. Most of the fundamental irregular ring even edge flexagons described in this chapter have a cycle in which main positions are flat non overlapping irregular even edge rings. There are numerous possible fundamental irregular ring even edge flexagons, most of which are difficult to handle, so only a selection is described. Most of these are reasonably easy to handle. Degenerate versions are usually easier to handle and examples are included.KeywordsFlexagonsIrregular RingRing EdgeMain PositionsNet EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.