The wizzle Butterfly belongs to the cmm symmetry group of the 17 wallpaper symmetry groups, which has reflection in two perpendicular directions and glide-reflection in two perpendicular directions in between. The axes of glide reflection are midlines of the reflection axes. It also has centers of 2-fold rotation (180°) located at the intersection of the reflection axes or the intersection of the glide reflection axes (Baloglou, 2007). Butterfly creates a periodic tessellation based on a square grid, thus possessing the trivial isometry of a 360° rotation as well as many trivial isometries of translation (horizontal, vertical, diagonal). In the above images, where you see the tessellation created by the Butterfly, you can easily discern the reflection axes and deduce the axes of glide reflection (midlines of the reflection axes) and the centers of double rotation (intersection points of the reflection axes or intersection points of the glide-reflection axes). These isometries constitute the elementary solving rule of the Butterfly. A set of four tiles creates the basic structural unit of the infinitely repeating (with translational isometry) pattern that fills the Euclidean plane and represents a second approach to solving the Butterfly.

For educational purposes, and not only, you can use the tiles of the Butterfly  to create all kinds of isometries (translation, reflection, glide-reflection, all types of rotations: 2-fold, 3-fold, 4-fold, 6-fold, etc., and compositions of isometries). However, not all of these isometries create a tessellation on the plane. You can find more details in the section "Wizzle in education".

Δείτε τη λύση του wizzle.