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 (12381492). 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.
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.


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.


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? 