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Math in Art

Splendid Symmetry
Syria<br><i>Hexagonal Wall Tile</i>, 16th century<br>Earthenware with turquoise and blue underglaze<br>Minneapolis Institute of Arts<br>The William Hood Dunwoody Fund
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Hexagonal Wall Tile, 16th century
Earthenware with turquoise and blue underglaze
Minneapolis Institute of Arts
The William Hood Dunwoody Fund


From the earliest times, artists have made use of symmetry and pattern in creating and decorating works of art and architecture. Symmetry involves specific ways of repeating a shape. There are several types of symmetry—reflection, rotation, and translation. In reflection symmetry, a shape is “flipped” across an imaginary line to produce its mirror image. In rotation symmetry, a shape is turned at a constant angle and repeated. In translation symmetry, a shape is repeated by sliding it (forward, backward, sideways), without flipping or turning. In all of these transformations, the shape’s line lengths and angles stay the same.

What kind of symmetry does this tile have? To answer, you need to know why the tile was made and where it belonged. Hexagonal (six-sided) tiles were used in Islamic countries to decorate mosques and palaces, both inside and out. In Islamic religious art, abstracted vines, leaves, and flowers became common motifs. Many artists depicted flowers and plants native to their own region. Often whole walls were covered with identical tiles that formed a continuous pattern. Repeating a single tile design over and over in all directions is an example of translation symmetry. (This particular tile also has reflection symmetry, since you can divide it into mirror-image halves.)

In designing decoration of all kinds, Islamic artists used symmetry for cultural reasons. The repetition of identical motifs symbolized unity in multiplicity, an important idea in Islam. The colors and plants chosen for this tile reflect the surroundings, interests, and artistic taste of the people living when and where the tile was made.

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1. This quilt, made by Amish women in Pennsylvania, is an example of reflection symmetry.
United States, Diamond in the Square Quilt, about 1910, wool and cotton, Minneapolis Institute of Arts, gift of Katherine Komanoff Goodman
2. In this example of rotation symmetry, the design in one-fourth of the light fixture is repeated three times, turning at 90-degree angles around the center.
William Gray Purcell, George Grant Elmslie, Mosaic Art Shops (E. L. Sharretts), Ceiling light fixture, 1913, glass, zinc caming, Minneapolis Institute of Arts, bequest of Anson Cutts, Jr.
3. The animal decoration of this Guatemalan weaving has translation symmetry. The design repeats as though sliding up and down and sideways, but it is not flipped or turned.
Maya culture, Guatemala, Tzute, 1935-40, cotton, Minneapolis Institute of Arts, gift of Richard L. Simmons


March 2010