Abstract
The general definition that a deformation twin is a region of a crystal which has sheared so that the orientation has changed but the structure is preserved gives rise to some twinning modes which do not obey the classical laws of deformation twinning. For example in the study of double twinning new types of twinning modes with four irrational elements and new orientation relationships arise in all crystal systems other than triclinic. In the double twinning mechanisms discussed, the simple equivalent modes which are of this type do not give the twin structure directly but involve atomic shuffling. However by considering the shearing of a square lattice into itself but with a different orientation it is shown that it is possible for this new type of mode to arise in which all lattice sites shear to their correct twin positions. This study also indicates that nominally compound twinning modes with four rational elements and complex shuffle mechanisms need in fact involve no shuffling if orientation relationships, other than those generally assumed, are permitted.