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:
          FFFFFF
A frieze pattern always has translational symmetry lengthwise down the band. In the pattern above, the letter F 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:

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Here the letter F 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 F's -- one forward, one backward:
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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:
          EEEEEE
It might help you see the symmetry if we draw guidelines on the pattern.
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Instead of reflection symmetry across the center of the band, a frieze pattern might have glide reflection symmetry:
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or halfturn symmetry:
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