#
Frieze Patterns

##
Lengthwise Symmetry

A *frieze pattern* is a pattern that has symmetry in a line. It looks
like a band of a repeated design. For example, the following pattern is
a frieze pattern:

A frieze pattern always has **translational** symmetry lengthwise down
the band. In the pattern above, the letter
is repeated over and over down the length of the band.
If the design is flipped each time it is repeated down the band, then
the pattern also has **reflection** symmetry. For example, this pattern
has reflection symmetry down the the length of the band:

Here the letter
is flipped each time it is repeated. Notice that the pattern still has
translational symmetry, if we consider the design being repeated as two
letter 's -- one
forward, one backward:

##
Crosswise Symmetry

Frieze patterns may have symmetries across the center of the band as well.
For example, this pattern has **reflection** symmetry across the center
of the band:

It might help you see the symmetry if we draw guidelines on the pattern.

Instead of reflection symmetry across the center of the band, a frieze
pattern might have **glide reflection** symmetry:

or **halfturn** symmetry:

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