# 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: