For the common “strip pattern” made of repeating shapes in a horizontal band, the transformations that map the strip pattern onto itself are:

  • A horizontal translation by the repeat distance of the pattern (sliding the whole strip left or right by exactly one “tile” length).
  • A reflection across a vertical line placed through the middle of a repeating unit, if each unit is symmetric that way.
  • A 180° rotation about the center of a repeating unit, when the motif has half‐turn symmetry.
  • A glide reflection (a reflection across a horizontal axis of the strip combined with a translation along the strip) when the motif alternates in such a way that flipping and sliding lines it back up.

In many textbook “strip pattern” problems, the intended full symmetry set is: horizontal translations, reflection across a vertical line, glide reflections, and 180° rotations, all of which carry the infinite strip back onto itself.