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Four Types of Repetition

The five isometries above are the only isometries that map Lonto itself. We will refer to these

isometries as I, R, T, H, V and G respectively.

Since these are the only isometries that map Lonto itself, frieze patterns must be generated by

repeating combinations of these actions. Repeating the action T indefinitely, results in a frieze

pattern that we call Pattern 1

Frieze Pattern 1

Simularly repeating the action G indefinitely also results in a frieze pattern, Pattern 2

Frieze Pattern 2

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In this pattern the tile is

but the pattern is generated from a single R by the repeated action of a glide reflection. This

samll piece of the pattern that has no symmetry and generates the entire design by

applications of I, R, T, H, V, and G is called a cell.

Repeating any of the other actions does not generate a strip required to be a frieze design. To

generate other patterns we are going to have to combine Pattern 1 or Pattern 2 with the other

actions. Again this can be done constructively on transparencies. Produce two copies each of

Patterns 1 and 2 and, in each case, place one copy on top of the other and then apply one of

the other actions on the top transparency.

For example applying the action V to Pattern 1 generates the pattern

Frieze Pattern 3

which we call Pattern 3. As above the vertical reflection is in the dashed green line.

Simularly applying action R by rotating Pattern 1 by 180oaround the green dot yieldsPattern 4.

Frieze Pattern 4

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Applying the rotation R to Pattern 2 gives rise to Pattern 5

Frieze Pattern 5

and the horizontal reflection H applied to Pattern 1 gives Pattern 6.

Frieze Pattern 6

Verify that no additional patterns result from applying any of the five isometries

to Pattern 1 or 2.

We now have to ask if additional patterns arise from applying any of the five isometries to

Patterns 3, 4, 5 or 6. Some experimentation shows that the horizontal reflection H applied to

Pattern 3 yields Pattern 7.

Frieze Pattern 7

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FOUR TYPES OF REPETITION

No. Types Directions Examples of Transformations

1. Translations

Repeating Motif Slides Up or

Down

Either

Vertical, Horizontal or Diagonal

2. Rotations

When motif turns around a

point.

Rotation can be

60,

90,

120

or

180

Order 6 in 60o

Rotation

Oder 4 in 90 Rotation

Oder 3 in 120 Rotation

Oder 2 in 180 Rotation

Vertical, Horizontal, Diagonal

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3. Reflections

When motif reflects and the

image reverses as in a mirror.

Vertical

4. Glide Reflections When motif translates along the

axis and at the same time

reflects across an axis.

Vertical Reflection

Vertical & Horizontal

Glide Reflection

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NO STRUCTURE TYPES

1 11

2 12

3 1g

4 m1

51m

6 mg

7 mm

Frieze pattern types and structures

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Frieze Patterns (Types & Structures)