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Geometry Session 7, Part C: Translation Symmetry and Frieze Patterns
Session 7 Part A Part B Part C Homework
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Session 7, Part C:
Translation Symmetry and Frieze Patterns

In This Part: Translation Symmetry | Frieze Patterns | Classifying Frieze Patterns

An infinite strip with a symmetric pattern is called a frieze pattern. There are only seven possible frieze patterns if we are using only one color. Note 3

1. Translation symmetry only:

2. Glide reflection plus translation symmetry:

3. Reflection over a horizontal line plus translation:

4. Reflection over a vertical line plus translation:

5. Rotation (a half-turn about a point on the midline of the strip) plus translation:

6. Reflection over a vertical line plus a reflection over a horizontal line plus translation:

7. Reflection over a vertical line plus glide reflection plus translation:

Problem C1


It may not be obvious how an infinite frieze pattern can be created from a basic element. Follow these step-by-step instructions to create a frieze using Design 6 (reflection over a vertical line plus reflection over a horizontal line plus translation). The instructions use the letter p as a basic design element of the pattern. The printable design element page (PDF) contains several versions of a more complex design element. Print this page and cut out the design elements. Then create the frieze pattern using this design element. Alternatively, you can draw your own design element to create the frieze pattern.

Step 1: Start with a basic design element. It's best if it is a nonsymmetric design so that the symmetry created by the transformations is more apparent.

Step 2: All frieze patterns have translation symmetry, so we'll leave that for last. Once we create a basic unit that contains all of our required other symmetries, we can translate it infinitely in both directions. So in this case, we'll start with a vertical reflection. Take your basic design element and reflect it over a vertical line. It's best to choose a line that is close to, but not intersecting, your original element.

There are now two pieces to your basic design: the original element and its reflected image.

Step 3: The next symmetry is horizontal reflection, so take your basic design block (now two elements) and reflect them both over a horizontal line. Again, choose a line that is close to, but not intersecting, your original design.

Step 4: We now have a basic element with all of the required symmetries except for translation. Take your basic element and translate it by a fixed distance in both directions. You have created a frieze pattern!

The seven patterns given are certainly not all the possible combinations of transformations. How can they be the only possible frieze patterns? It turns out that other combinations fall into one of these categories as well. That is, they create equivalent patterns.


Problem C2


Use the printable design element, or draw your own design element, to create the seven frieze patterns.


Problem C3


Frieze patterns appear in the artwork of Native American and African cultures, as well as in cornices on buildings. Create a more interesting basic design element, and create a frieze pattern with that element. Choose one of the seven patterns described above.


Problems in Part C adapted from the NCTM Addenda Series, developed by North Central Regional Education Laboratory. pp. 34-35, 42. © 1991 by National Council of Teachers of Mathematics. Used with permission. All rights reserved.

Next > Part C (Continued): Classifying Frieze Patterns

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