Frieze Patterns
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Transcript of Frieze Patterns
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