# Cross-Check Patterns

By applying the 24 rotation symmetries to the corner facelets of the cube one may generate the Cross Pretty Pattern Group. These patterns may be arranged into five conjugate classes: the identity cube, six order two 6-cross patterns, eight order 3 6-cross patterns, six order 4 4-cross patterns and three order 2 4-cross patterns.

By applying the 24 Th symmetries to the edge facelets of the cube one may generate the Check (or Checkerboard ) Pretty Pattern Group. These patterns may be arranged into six conjugate classes: the identity cube, pons asinorum, eight order three 6-check patterns, eight order six 6-check patterns, three order two 4-check patterns and three order two 2-check patterns.

The product of these two groups is the Cross-Check Pattern Group to coin a term. The 576 elements of this group have various combinations of plaid, cross, check and spot patterns on the cube faces. Symmetry reduces this group to 46 basic patterns (45 minus the identity cube). I have put together a table describing these patterns with turn sequences for their generation. Those interested in pretty patterns may find the table in the Rubik section of my web site.