Symmetric Patterns at the Alhambra

With Math Problems

History

Spain was conquered in 714 A.D. by Muslim armies (after being conquered by, among others, Romans and Visigoths). During the 800 years until Spain was reconquered by Christians, the Muslims greatly influenced the culture of Spain. During the middle ages, when little mathematics and science were being done in the rest of Europe, Spain was an intellectual center.

The Alhambra is a walled city and fortress in Granada, Spain. It was built during the last Islamic sultanate on the Iberian peninsula, the Nasrid Dynasty (1238-1492). The palace is lavishly decorated with stone and wood carvings and tile patterns on most of the ceilings, walls, and floors. Islamic art does not use representations of living beings, but heavily uses geometric patterns, especially symmetric (repeating) patterns.

Technical note

Photos on this page were taken with an Apple Quick Take digital camera, fixed up with Adobe Photoshop, and transferred by ftp from Spain to the Math Forum in Pennsylvania.

The Patterns and the Math

For mathematical background about symmetry see The Four Plane Symmetries. A wallpaper pattern is one that has translation symmetry in two directions (such as left/right and up/down). A frieze, or strip, pattern is one that has translation symmetry in one direction. A rosette pattern is one that has no translation symmetry, just reflection and/or rotation symmetry.

Can you find all the symmetries in each of the patterns shown below?
 

Pattern 1 This pattern occurs as a frieze pattern, but it can be extended to a wallpaper pattern covering the whole plane. Try drawing the pattern with straightedge and compass, a dynamic geomerty package such as Cabri or Geometer's Sketchpad, or on isometric dot paper (dots arranged in a pattern of equilateral triangles).
Pattern 2 This wallpaper pattern has rotational symmetry (by what angles?) Can you find the centers of rotation? Try drawing the pattern on square graph paper.
Another variation on pattern 2
Pattern 3 This pattern is made with glaze on square tiles; the tiles are then lined up in a frieze pattern. 
  • What symmetries does a single tile have? 
  • What symmetries does the frieze pattern have? (Theoretically, a frieze pattern goes on forever; 3 tiles of the frieze are shown in this picture). 
  • If the tiles covered the whole plane to make a wallpaper pattern, what symmetries would it have? 
Pattern 4 This pattern is also made with glazed square tiles, forming a wallpaper pattern. What symmetries does it have? Are they the same type as pattern 3? 
Pattern 5 This wallpaper pattern is made from a single tile shape. (NOT square!) Pay no attention to the fact that the pattern makes a detour to cover a column. What are the symmetries of the wallpaper pattern?
Pattern 6 This frieze pattern is a border at the top of pattern 5. 
  • What are its symmetries? 
  • This pattern is a good example of color symmetry : each symmetry of the pattern (considered as just a symmetry of shapes) either takes all the blacks to blacks and all whites to whites, or switches all the blacks with the whites. Which symmetries change the colors? Which symmetries keep the colors the same? 
Pattern 7 Does this pattern have reflection symmetry? Rotation? If so, by what angles? You can make a copy of this pattern on isometric dot paper.
Pattern 8 This is a carved wood ceiling panel. Imagine that it goes on forever as a wallpaper pattern. Find all the lines of reflection symmetry. What sorts of triangles do these lines form?
Photographs by Susan Addington and David Marshall, The California Math Show.