The wizzle Geo 5 belongs to the pgg symmetry group of the 17 wallpaper symmetry groups, which has glide-reflection in two perpendicular directions, but lacks reflection. It also has two pure 2-fold rotation centers (180°) but these are not on the glide-reflection axis (Baloglou, 2007). It 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 Geo 5, you can observe the distinct centers of 2-fold rotation in knobs of the square grid and in between them the glide reflection axes. These isometries also constitute the elementary solving rule of the Geo 55. A set of four tiles forms a meta-tile, the basic structural unit of the infinitely repeating (with translation isometry) pattern that fills the Euclidean plane, providing a second approach to solving the Geo 5.

For educational purposes, and not only, you can use the tiles of the Geo 5 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.