The concept of Coincidence-Site Lattice (CSL) is generalized theoretically by studying a case in which an ice crystal is cut along a crystallographic plane {h_1h_2h_3k} and the upper half of its crystal lattice is rotated by an angle of 180° about an axis perpendicular to {h_1h_2h_3k} at a lattice point chosen as the origin. After the rotation, all the lattice points of the upper half along the composition plane coincide exactly with their counterpart points of the lower half, thus composing the parallerogrammic CSL with a value of Σ=1, while the bonding at each coincidence site is not always perfect. Because of this, the possibility of formation of twins whose twin axes are perpendicular to {h_1h_2h_3k} is accounted for by three factors: (i) λ, the reciprocal density of coincidence sites, (ii) β, the angle between a bond arm and its counterpart bond arm facing each other at the coincidence sites, and (iii) the number of bond arms accomplished at a coincidence site. Possible twinned structures expected in ice crystals are twinning relations with {3034}, {3038}, {3032}, {1011}, {1012}, and so on, as composition planes. On the basis of the above theoretical findings, an explanation was made as to the snow crystal structures of a combination of bullets and prisms as well as a spatial assemblage of plate crystals observed in nature.
