1.3: Affine Isomorphism
Each symmetry operation of crystallographic group in E 3 may be represented by a 3×3 matrix W (the linear part ) and a vector w . Two crystallographic groups G 1 = {( W 1 i , w 1 i )} and G 2 = {( W 2 i , w 2 i )} are called affine isomorphic is there exists a non-singular 3×3 matrix A and a vector a such that:
G 2 = {( A , a )( W 1 i , w 1 i )( A , a ) -1 }
Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an affine mapping of E 3 . Two affine isomorphic groups are always isomorphic.