Frieze Patterns

A frieze is a pattern which repeats in one direction. Friezes are often seen as ornaments in architecture. A mathematical analysis reveals that there are seven different frieze patterns possible.

Pattern 1: Translation

The only transformation in this pattern is a translation.

Pattern 2: Glide reflection

The transformation in this pattern is a glide reflection. All frieze patterns also map onto themslves under a translation.

Pattern 3: Two parallel reflections

The transformations in this pattern are two parallel reflections across vertical mirrors..

Pattern 4: Two half turns

The transformations in this pattern are two half turns.

Pattern 5: A reflection and a half turn

The transformations in this pattern are a reflection across a vertical mirror and a half turn.

Pattern 6: Horizontal reflection

The transformations in this pattern are a translation and a reflection across a horizontal mirror.

Pattern 7: Three reflections

The transformations in this pattern are three reflections, one across a horizontal mirror and the other two across parallel vertical mirrors.

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:           

Frieze GroupsLengthwise, a frieze pattern may have reflection symmetry or just translation symmetry.

Crosswise, it may have reflection symmetry, glide reflection symmetry, halfturn symmetry or no symmetry.

There are many different combinations of these symmetries that can be found in frieze patterns. For example, it might have translation symmetry lengthwise and reflection symmetry crosswise.

Or, it could have reflection symmetry lengthwise and no symmetry crosswise. Or, it could have some other combination of lengthwise and crosswise symmetry.

Each possible combination of symmetries is called a frieze group. It turns out that there are 7 different unique frieze groups. (We can prove this, but that's another exercise.)

     Hop       

     Spinhop   

     Jump      

     Sidle     

     Step      

     Spinjump  

     Spinsidle 

The exercises on the next two pages will help you figure out which frieze groups have which symmetries.

Frieze Groups

Lengthwise Symmetry

Looking at patterns in the different frieze groups, figure out which type(s) of lengthwise symmetry each frieze group has.

     Hop       

           translation    reflection

     Spinhop   

           translation    reflection

     Jump      

           translation    reflection

     Sidle     

           translation    reflection

     Step      

           translation    reflection

     Spinjump  

           translation    reflection

     Spinsidle 

           translation    reflection

Frieze Groups

Crosswise Symmetry

Looking at patterns in different frieze groups, figure out which type(s) of crosswise symmetry each frieze group has.

Guide lines across the center of the band are only drawn for the first two groups. It may help you understand the symmetries to think about or actually draw guide lines for the other cases.

     Jump      

      reflection    glide reflection    halfturn    none

     Spinhop   

      reflection    glide reflection    halfturn    none

     Hop       

      reflection    glide reflection    halfturn    none

     Sidle     

      reflection    glide reflection    halfturn    none

     Step      

      reflection    glide reflection    halfturn    none

     Spinjump  

      reflection    glide reflection    halfturn    none

     Spinsidle 

      reflection    glide reflection    halfturn    none

Frieze Groups

Answer Key

Be sure to discuss the answers with the students so that they understand why some problems have more than one answer.

Lengthwise Symmetry

     Hop       

           translation    reflection

     Spinhop   

           translation    reflection

     Jump      

           translation    reflection

     Sidle     

           translation    reflection

     Step      

           translation    reflection

     Spinjump  

           translation    reflection

     Spinsidle 

           translation    reflection

Crosswise Symmetry

     Jump      

      reflection    glide reflection    halfturn    none

     Spinhop   

      reflection    glide reflection    halfturn    none

     Hop       

      reflection    glide reflection    halfturn    none

     Sidle     

      reflection    glide reflection    halfturn    none

     Step      

      reflection    glide reflection    halfturn    none

     Spinjump  

      reflection    glide reflection    halfturn    none

     Spinsidle 

      reflection    glide reflection    halfturn    none