Four dimensional space is a purely theoretical idea but is nevertheless fascinating. Alicia Boole Stott (1860–1940) was renowned for her ability to visualise four dimensional geometric objects (Coxeter 1963). Most people find this difficult, but it is worth the effort. As an example of the interesting results that can be obtained the dynamic behaviour of Rubik's tesseract, the four dimensional analogue of Rubik's cube, has been investigated (Velleman 1992).

This chapter is a brief introduction to the remarkably rich and largely unexplored topic of ‘flexahedra’, which are the four dimensional analogues of flexagons. The nets of flexahedra are three dimensional so can be visualised in ordinary space. Sometimes main and intermediate positions of flexahedra are also three dimensional and hence may be visualised. It is possible to generate a flexahedron analogue of any flexagon, and examples are given. The dynamic behaviour of the flexahedron generated is analogous to that of the initial flexagon. There are some flexahedra which are not analogues of flexagons, and one is described. It is of course not possible to make physical models of flexahedra.

Most of the material in this chapter was developed about 30 years ago but has not previously been published. Examples have been chosen to make visualisation as easy as possible.