When I learned about the operations of symmetry, I also learned about a notation system used by crystallographers and chemists to describe 1-, 2-, and 3-dimensional patterns. I will use this shorthand to describe eleven of the seventeen planar symmetry groups, also known as wallpaper symmetries. The remaining six symmetry groups pertain to triangular motifs and are not included in this discussion.
Group p1: translates the tile. The double-headed arrows indicate which directions the tile can move.
Planar Symmetry Group p1
Group p2/I: translates a pair of rotated tiles. The double-headed arrows indicate which directions the tiles can move; the circles represent centers of rotation.
Planar Symmetry Group p2/I
Group p2/II: translates a pair of rotated tiles. The double-headed arrows indicate which directions the tiles can move; the circles represent centers of rotation.
Planar Symmetry Group p2/II
Group p2/III: rotates the tile along all sides. The circles represent centers of rotation.
Planar Symmetry Group p2/III
Group p4: rotates the tile around one of its corners. The circle represents the center of rotation.
Planar Symmetry Group p4
Group pm/I: translates a pair of reflected tiles. The double-headed arrows indicate which directions the tiles can move; the solid line represents a mirror.
Planar Symmetry Group pm/I
Group pm/II: translates a pair of reflected tiles. The double-headed arrows indicate which directions the tiles can move; the solid line represents a mirror.
Planar Symmetry Group pm/II
Group p2mm: reflects the tile along all sides. The solid lines represent mirrors.
Planar Symmetry Group p2mm
Group cm/I: reflects a pair of glide-reflected tiles. The solid line represents a mirror; the dashed lines represent mirror/translation lines.
Planar Symmetry Group cm/I
Group cm/II: reflects a pair of glide-reflected tiles. The solid line represents a mirror; the dashed lines represent mirror/translation lines.
Planar Symmetry Group cm/II
Group p2mg/I: reflects a pair of rotated tiles. The solid line represents a mirror; the circles represent centers of rotation.
Planar Symmetry Group p2mg/I
Group p2mg/II: reflects a pair of rotated tiles. The solid line represents a mirror; the circles represent centers of rotation.
Planar Symmetry Group p2mg/II
Group c2mm/I: rotates pairs of reflected tiles. The solid lines represent mirrors; the circles represent centers of rotation.
Planar Symmetry Group c2mm/I
Group c2mm/II: rotates pairs of reflected tiles. The solid lines represent mirrors; the circles represent centers of rotation.
Planar Symmetry Group c2mm/II
Group p4gm: reflects a Group p4 block on all sides. The circles represent centers of rotation; the solid lines represent mirrors.
Planar Symmetry Group p4gm
Group pg/I: translates a pair of glide-reflected tiles. The double-headed arrows indicate which directions the tiles can move; the dashed lines represent mirror/translation lines.
Planar Symmetry Group pg/I
Group pg/II: translates a pair of glide-reflected tiles. The double-headed arrows indicate which directions the tiles can move; the dashed lines represent mirror/translation lines.
Planar Symmetry Group pg/II
Group p2gg: glide-reflects the tile along all sides. The dashed lines represent mirror/translation lines.
Planar Symmetry Group p2gg
