Complex Flexagons

Abstract

The trihexaflexagon (Section 4.2.3), in which one cycle can be traversed, was discovered by Stone in 1939. He immediately realised that more complicated flexagons were possible, and discovered the hexahexaflexagon, in which four cycles can be traversed (Conrad and Hartline 1962; Pook 2003). The hexahexaflexagon is an example of a complex flexagon.

A complex flexagon consists of two or more solitary flexagons (Section 4.2.1). Its dynamic properties include features of the dynamic properties of the precursor flexagons. The torsion of a complex flexagon is the algebraic sum of the torsions of the constituent flexagons. Complex flexagons can also incorporate parts of solitary flexagons. The characteristic flex for a complex flexagon is the same as that for the precursor flexagons. Most of the more interesting flexagons for which nets have been published are complex flexagons, and include some spectacular examples. For this reason it would have been better to have introduced the concept of a complex flexagon earlier in the book. However, material on solitary flexagons in Chapters 4–10 is needed as a preliminary to the discussion of complex flexagons. Solitary flexagons are broadly equivalent to single polyhedra, whereas complex flexagons are broadly equivalent to compound polyhedra, such as the well known stella octangula, which is a compound of two regular tetrahedra (Fig. 1.14, Cromwell 1997). There are several ways in which two solitary flexagons, with the same type of leaf, can be joined together to form a complex flexagon.