Wallpaper Exercises

1. Print out the six wallpaper patterns in File:Wall-exercise-patterns.pdf.

   1. For patterns A,B,C, mark all reflection and glide reflection symmetries.
   2. For patterns D,E,F, mark all centers of rotation symmetry.




2. Decide which of the 17 symmetry groups each of these Escher drawings has. You can ignore colors.
   1. Sketch #6 (Camels)
   2. Sketch #13 (Dragonflies)
   3. Sketch #21 (Men)
   4. Sketch #67 (Horsemen)
   5. Sketch #69 (Birds, fish, turtles)
   6. Sketch #105 (Pegasus)
   7. Sketch #119 (Fish)
3. The Cosmati were a family of artisans in Medieval Rome who laid beautiful tile floors throughout the city, notably in many of the churches of the period. The examples below are pictures taken at Santa Maria in Cosmedin. Find the symmetry groups of these tilings, and briefly explain how you dealt with the colors of the tiles.
   (Click on the pictures for larger versions.)
   • Cosmati Tiles at Santa Maria in Cosmedin
   • 
     Pattern A
   • 
     Pattern B
   • 
     Pattern C
   • 
     Pattern D




4. The ancient Egyptians decorated their tombs with some interesting patterns. Describe the symmetries of these patterns. (Click on the pictures for larger versions.)
   
   • Ceiling decorations in Egyptian Tombs
   • 
     Pattern A
   • 
     Pattern B
   • 
     Pattern C
   • 
     Pattern D




5. Identify the symmetry group of each pattern in File:Wall-exercise-patterns.pdf



G. Polya, Patterns illustrating the 17 wallpaper groups. Click for larger images.

6. George Pólya came up with his own names for the 17 wallpaper groups. Pólya's picture, with the names, is shown at right. See also page 23 of Visions of Symmetry and Escher's sketches on the following pages. Figure out the IUC names for his 17 patterns, and make a chart showing Pólya's names in one column and the corresponding IUC names in the other column. For example, Pólya's $D_{4}^{*}$ is p4m in IUC notation.
   (Note that Pólya's $D_{2}kkkk$ is colored, and he considers the color preserving symmetry group)
7. Can a wallpaper pattern contain:
   1. An order 4 rotation and an order 6 rotation?
   2. An order 4 rotation and an order 3 rotation?
   3. An order 3 rotation and an order 6 rotation?
   4. An order 2 rotation and an order 3 rotation?
   5. Perpendicular mirror lines and an order 6 rotation?
   6. Perpendicular mirror lines and an order 3 rotation?
8. Sketch an interesting pattern with symmetry group p4m.
9. Find four different patterns used for laying bricks (look around). Sketch them on graph paper, and decide which symmetry group each one has.
10. In Frieze Exercises#motif, you picked a motif and then used the motif to create seven frieze patterns, one with each possible symmetry group. Use the letter as a motif to create wallpaper patterns as follows:
    
    1. The symmetry group p1 has only translations. Create a p1 pattern using the letter P.
    2. The symmetry group pm has translations and reflections. Create a pm pattern using the letter P.
    3. The symmetry group p4 has only translations and 4-fold rotations. Create a p4 pattern using the letter P.
    4. Pick one other symmetry group and create a pattern with that symmetry group, using the letter P.

Instructor:Wallpaper Exercises Solutions (Instructors only).