A border pattern, or frieze pattern, is a design obtained from a motif by repeating that motif infinitely often in two opposing directions along a strip. Border patterns are traditionally oriented horizontally, thus the patterns are understood to continue both to the left and right without end. A border pattern falls into one of seven symmetry types. This Demonstration provides one example of each type to let you explore the effects of reflection, rotation, and translation on those patterns.
When you select the "show highlighted motif" option, one motif appears in blue and the one immediately to its left appears in red. This lets you see the action of a reflection, rotation, or translation more easily. The axes of reflection used go through the center point of the blue motif, and that point is the center of the rotations as well. Note that in the case of a border pattern of type mg the vertical axis over which the border pattern has symmetry is not through the center of the motif (see Snapshot 5).
All border patterns have translational symmetry. Border patterns are classified according to which other symmetries they exhibit. The seven symmetry types of a border pattern are:
11: no other symmetry
m1: reflection over a vertical axis, but not a horizontal axis
1m: reflection over a horizontal axis, but not a vertical axis
12: rotation by 180 degrees, but no reflection symmetry
mm: reflection over both a horizontal axis and a vertical axis (which implies rotational symmetry)
1g: glide reflection, but no reflection
mg: reflection over a vertical axis and glide reflection (which implies rotational symmetry)