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 the seven linear symmetry groups, also known as frieze symmetries.
Group t: translates the tile. The double-headed arrows indicate which directions the tile can move.
Linear Symmetry Group t
Group t2: translates a pair of rotated tiles. The double-headed arrows indicate which directions the tile can move; the circles represent centers of rotation.
Linear Symmetry Group t2
Group tm: reflects the tile. The solid lines represent mirrors.
Linear Symmetry Group tm
Group t2mm: reflects the tile horizontally and vertically; diagonal tiles rotate 180 degrees. The solid lines represent mirrors; the circles represent centers of rotation.
Linear Symmetry Group t2mm
Group mt: translates a pair of reflected tiles. The double-headed arrows indicate which directions the tile can move; the solid line represents a mirror.
Linear Symmetry Group mt
Group t2mg: reflects a pair of rotated tiles. The solid lines represent mirrors; the circles represent centers of rotation.
Linear Symmetry Group t2mg
Group tg: glide-reflects the tile. The dashed line represents the mirror/translation line.
Linear Symmetry Group tg
