## Learning Math: Geometry

# Symmetry Part C: Translation Symmetry and Frieze Patterns (60 minutes)

## Session 7, Part C

### In this part

- Translation Symmetry
- Frieze Patterns
- Classifying Frieze Patterns

### Translation Symmetry and Frieze Patterns

A translation or slide involves moving a figure in a specific direction for a specific distance. Vectors are often used to denote translation, because the vector communicates both a distance (its length) and a direction (the way it is pointing).

The vector shows both the length and direction of the translation.

A glide reflection is a combination of two transformations: a reflection over a line, followed by a translation in the same direction as the line.

Reflect over the line shown; then translate parallel to that line.

Only an infinite strip can have translation symmetry or glide reflection symmetry. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself. For glide reflection symmetry, you can reflect the pattern over some line, then slide in the direction of that line, and it looks unchanged.

The patterns must go on forever in both directions.

### 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.

### Classifying Frieze Patterns

Note that the point of the following problems is not to memorize the classification system, but rather to be able to make sense of this kind of a system, interpret what the symbols mean, and apply it to new situations.

Mathematicians have developed a two-character notation to denote each possible frieze pattern.

- The first character is m or 1 according to whether there is reflection over a vertical line or not.
- The second character is m if there is a reflection over a horizontal line, g if there is a glide reflection, 2 if there is a rotation, and 1 if none of these exist.

#### TAKE IT FURTHER

**Problem C4: Take it Further**

Classify each of the seven frieze patterns from the the section above according to this system.

**Problem C5: Take it Further**

The symbol m2 was not part of the classification you used. (It was the only combination of the two symbols missing.)

a. |
What would m2 represent in terms of symmetries? |

b. |
Create a simple frieze pattern with m2 symmetry. |

c. |
What is another name for the symmetry in your pattern? |

**Problem C6**

Using the codes for frieze patterns, classify each of the following designs. Assume the designs are really infinite.

### Notes

**Note 3**

If you are working with a group, consider going through these patterns one by one on a large poster. Have participants come up and show the lines and points in question. As you introduce the seven frieze patterns, work through Problem C1 as a whole group.

### Solutions

**Problem C1**

**Step 1:**

**Step 2:**

**Step 3:**

**Step 4:**

**Problem C2**

Using a new design element, the seven patterns are as follows:

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 reflection over a horizontal line plus translation:

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

**Problem C3**

Answers will vary depending on the choice of the basic element as well as of the pattern.

**Problem C4**

Following the order in which the patterns are originally described, they can be classified as 11, 1g, 1m, m1, 12, mm, mg.

**Problem C5**

a. |
It would be a combination of translation symmetry (which is present in all patterns), followed by a reflection about a vertical line (hence the m), followed by the 180° rotation about a point on the midline (hence the 2). |

b. |
See picture: |

c. |
The symmetry described is equivalent to pattern #7, so it can also be described as having translation symmetry, glide reflection, and reflection over a vertical line. So it could also be classified as mg. |

**Problem C6**

If you consider symmetry in the basic design element part of the symmetry of the frieze pattern, you may have slightly different answers than the ones here. In the order in which the designs are given, the classifications are as follows: 12, 11 (or possibly m1), m1, 11 (or possibly 1m), 11, 11 (or possibly m1), 11 (or possibly 1m), 11 (or possibly mm), 12, m1 (or possibly mm), m1 (or possibly mm), and 11 (or possibly 1m).