Fundamental Nets

Abstract

The nets used to construct flexagons are usually strips of hinged convex polygons. If the polygons are regular and connected by edge hinges, then nets can be completely defined as a sequence of hinge angles. The net used of course determines the appearance and dynamic properties of a flexagon. Some nets can be assembled to produce more than one distinct type of flexagon; this means that to specify a flexagon completely both the net and the method of assembly has to be specified. All flexagons exist as enantiomorphic (mirror image) pairs. The two members of an enantiomorphic pair of flexagons are not usually regarded as distinct types (Section 1.2).

The appearance of nets varies widely, and some are very irregular. Fundamental edge nets have a regular appearance and can be defined as a repeating sequence of hinge angles, without specifying the number of repetitions. In a first order fundamental edge net the hinge angles are all the same, but alternate hinge angles are of opposite sign. The hinge angles of a first order fundamental edge net are the same as the vertex angles of a regular polygon that is associated with the net. Fundamental flexagons also have associated polygons, and they are a key concept in flexagon theory.