Frieze as ClassSym

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© 2008 National Council of Teachers of Mathematicshttp://illuminations.nctm.org

Cyclic Figures — Notes NAME ___________________________

Two-dimensional figures that have only rotation symmetry are said to be cyclic. If a figure has two

rotation symmetries, it is classified as a Cyclic 2 figure. If it has three rotation symmetries, it is

classified as a Cyclic 3 figure. Shown below are examples of cyclic figures, their classification, and

a list of their rotation symmetries.

Figure Classification Rotation Symmetries

Cyclic 3 R 120, R 240, R 360

Cyclic 4 R___, R___, R___, R___

Cyclic ___ ______________________

______________________

Questions:

1. How do you determine the classification?

2. How do you find the angles of rotation?

Classify the following cyclic figures and list the rotation symmetries.

3. 4.




Dihedral Figures — Notes

Two-dimensional figures that have both rotation and reflection symmetry are said to be dihedral.

If a figure has two rotation symmetries and two reflection symmetries, it is classified as a Dihedral

2 figure. If it has three rotation symmetries and three reflection symmetries, it is classified as a

Dihedral 3 figure. Shown below are examples of dihedral figures and their classification.

Figure Reflection Lines Rotation Symmetries Classification

5 R 72, R 144, R 216, R 288, R 360 Dihedral 5

______R 60, R___, R___, R___,

R___, R 360

Dihedral 6

______ _________________ _________________

___________

Classify the following dihedral figures and sketch in the lines of reflection.

1. 2.




Seven Classes of Frieze Patterns — Notes

It has been proven that all infinitely repeating frieze patterns can be categorized into seven different

classes depending on the type(s) of symmetries the patterns contain. Shown below are the seven

classes of frieze patterns. Each figure gets transformed one, two, or three times to create the design.

Class I – Translation – Translate the original figure, and then repeatedly translate the new image.

Class II – Glide Reflection – Reflect and glide the original figure, and then repeat the glide

reflection on the newest image

Class III – Vertical Reflection – Reflect the original image on a vertical line, and then repeatedly

reflect the newest image on a new vertical line.

Class IV – 180° Rotation – Rotate the original figure 180° about a point and repeatedly rotate the

newest image to get the next image.

Class V – Vertical Reflection and then 180° Rotation – Reflect the original image along avertical line, and then repeatedly rotate the pair 180° about a point.

Class VI – Horizontal Reflection and then Translation – Reflect the original figure along a

horizontal line for one image and then translate the pair of images to initiate the pattern.

Class VII – One Horizontal and then Vertical Reflection – Apply a reflection along a horizontalline to obtain a second image. Repeatedly reflect the resulting pair along a vertical line to get the

pattern.




Classifying Frieze Patterns — Notes

A useful tool in classifying frieze patterns is creating a flowchart. Working with a partner using the

Seven Classes of Frieze Patterns activity sheet, answer the questions in the flowchart and fill in the

proper class number in each of the blanks.

Use the flowchart to classify the following frieze patterns:

1.

2.




Classifying Symmetry — Practice NAME ___________________________

1. Classify the following figures as either cyclic or dihedral and by the number of symmetries.

a) b)

c) d)

2. Hubcaps (wheel covers) are often either cyclic or dihedral figures. Classify the following:

a) b)

3. For each of the following, create the indicated figure from the initial pattern shown:

a) Cyclic 6 b) Dihedral 3




Classifying Frieze Patterns — Notes

A useful tool in classifying frieze patterns is creating a flowchart. Working with a partner using the

Seven Classes of Frieze Patterns activity sheet, answer the questions in the flowchart and fill in the

proper class number in each of the blanks.

Use the flowchart to classify the following frieze patterns:

1.

2.

Frieze as ClassSym

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