A frieze pattern or border pattern is a pattern that extends to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation. In addition to being mapped onto itself by a horizontal translation, some frieze patterns can be mapped onto themselves by other transformations.
1. Translations T
2. 180° rotation R
3. Reflection in a horizontal line H
4. Reflection in a vertical line V
5. Horizontal glide reflection G
Example 1 :
Describe the transformations that will map each frieze pattern onto itself.
Solution :
a. This frieze pattern can be mapped onto itself by a horizontal translation (T).
b. This frieze pattern can be mapped onto itself by a horizontal translation (T) or by a 180° rotation (R).
c. This frieze pattern can be mapped onto itself by a horizontal translation (T) or by a horizontal glide reflection (G).
d. This frieze pattern can be mapped onto itself by a horizontal translation (T) or by a reflection in a vertical line (V).
T - Translation :
TR - Translation and 180° rotation :
TG - Translation and horizontal glide reflection :
TV - Translation and vertical glide reflection :
THG - Translation, horizontal line reflection and vertical glide reflection :
TRVG - Translation, 180° rotation, vertical line reflection and horizontal glide reflection :
TRHVG - Translation, 180° rotation, horizontal line reflection, vertical line reflection and horizontal glide reflection :
To classify a frieze pattern into one of the seven categories, first decide whether the pattern has 180° rotation. If it does, then there are three possible classifications: TR, TRVG, and TRHVG.
If the frieze pattern does not have 180° rotation, then there are four possible classifications: T, TV, TG, and THG. Decide whether the pattern has a line of reflection. By a process of elimination, you will reach the correct classification.
Example 2 :
Categorize the snakeskin pattern of the mountain adder.
Solution :
This pattern is a TRHVG. The pattern can be mapped onto itself by a translation, a 180° rotation, a reflection in a horizontal line, a reflection in a vertical line, and a horizontal glide reflection.
Example (Identifying Frieze Patterns) :
The frieze patterns of ancient Doric buildings are located between the cornice and the architrave, as shown below.
The frieze patterns consist of alternating sections. Some sections contain a person or a symmetric design. Other sections have simple patterns of three or four vertical lines.
Portions of two frieze patterns are shown below. Classify the patterns.
Solution :
a. Following the diagrams on the previous page, you can see that this frieze pattern has rotational symmetry, line symmetry about a horizontal line and a vertical line, and that the pattern can be mapped onto itself by a glide reflection. So, the pattern can be classified as TRHVG.
b. The only transformation that maps this pattern onto itself is a translation. So, the pattern can be classified as T.
Example 3 :
A border on a bathroom wall is created using the decorative tile at the right. The border pattern is classified as TR. Draw one such pattern.
Solution :
Begin by rotating the given tile 180°. Use this tile and the original tile to create a pattern that has rotational symmetry. Then translate the pattern several times to create the frieze pattern.
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