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Frieze Patterns and Tessellations - Cycle 3
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 translations can be used.
Frieze pattern obtained by translation
In a frieze pattern produced by translation, the ensuing patterns are identical to the initial pattern. They have the same orientation, color arrangement, size, etc.
It gets easier to create new frieze patterns when this definition and this example are kept in mind.
Constructing a Frieze Pattern by Translation
Use the same approach as presented above.
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Trace the basic pattern
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Draw the frieze pattern’s translation arrow
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Perform the first translation of the basic pattern
Repeat the translation as often as desired
Regardless of the colours and basic patterns used, simply follow the steps to build a frieze pattern worthy of its name.
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 Translation
It's possible to use the same procedure to produce a tessellation by translation as when using a reflection. It's important to always make sure there are no empty spaces between the patterns.
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Trace the basic pattern
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Draw the translation arrow
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Sketch the result of the first translation
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Repeat the translations as often as desired
It's important to note that the orientation, direction, and length of the translation arrow will vary depending on the image desired. There can be no free space between any of the patterns. The surface must be completely covered.