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Frieze Patterns and Tessellations - Cycle 2
Concept sheet | Mathematics
Many fabrics and materials have repeated images or patterns with a pre-established order. Mathematicians refer to these as frieze patterns (friezes) and/or tessellations.
Geometric transformations are needed to produce a frieze pattern and/or tessellation.
Frieze Pattern (Definition and Properties)
In general, a frieze pattern is a repeating pattern that follows a particular order.
A frieze is a continuous strip with parallel edges, formed by repeating one or more patterns. The patterns repeat regularly and harmoniously.
To create the strip, successive reflections can be used.
Frieze pattern obtained by reflection
In a frieze pattern produced by reflection, each pattern is a reflection of one preceding it. Thus, the dimensions are preserved, but the pattern’s orientation changes.
It gets easier to create new frieze patterns when this definition and this example are kept in mind.
Constructing a Frieze Pattern by Reflection
As in many mathematical concepts, we can establish a sequential order that enables the frieze pattern to be properly constructed.
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Trace the basic pattern
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Draw the frieze pattern’s axis of reflection
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Perform the reflection for the pattern’s first section
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Perform a reflection on the rest of the pattern
Tessellation
It is equally important to take into account both the basic pattern and space it occupies.
A tessellation is a surface that is completely covered in patterns. It has no empty spaces and no overlap.
Constructing a Tessellation by Reflection
A tessellation can be obtained by reflecting one or several patterns.
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Trace the basic pattern
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Draw a line of reflection
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Perform the first reflection
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Reflect as often as desired
It's important to note that lines of reflection can be drawn anywhere. The entire surface must be covered without leaving any empty spaces between patterns.