Operations of Symmetry

There are four operations of symmetry: translation, rotation, reflection, and glide reflection. I present a brief explanation and example of each.

Translation: The motif moves horizontally, vertically, or diagonally, without changing orientation. In the examples below, the motif is a square with lines drawn through it; "T" indicates the top of the motif. The arrows indicate movement of the motif.

Rotation: The motif moves around a center of rotation, or rotocenter, along its sides or at its corners. In the examples below, the motif rotates 180 degrees at rotocenters (indicated by circles) along its sides, then rotates 90 degrees at a rotocenter (indicated by a circle) at its corners.

Reflection: The motif reflects horizontally or vertically along a mirror line. In the examples below, the dotted lines indicate a mirror; "M" indicates a mirror-image motif.

Glide Reflection: The motif is both translated and reflected. In the example below, the mirror lines run through the center of the motifs, horizontally and vertically. The motifs are translated vertically, while mirrored horizontally; they are translated horizontally, while mirrored vertically.