# 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.

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.)

4. The ancient Egyptians decorated their tombs with some interesting patterns. Describe the symmetries of these patterns. (Click on the pictures for larger versions.)

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

6. G. Polya, Patterns illustrating the 17 wallpaper groups.
7. 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)

8. 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?

9. Sketch an interesting pattern with symmetry group p4m.

10. Find four different patterns used for laying bricks (look around). Sketch them on graph paper, and decide which symmetry group each one has.

11. 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.

12. Look at all the sketches in Escher's regular division notebook. These are on pages 116-229 of Visions of Symmetry or at Regular Division of the Plane Drawings. Find ten sketches featuring four-legged mammals. (A four-legged mammal has four legs, and is a mammal - horse, dog, pegasus, lion, etc. etc. No people, fish, lizards.)
1. Find the wallpaper symmetry group for each of the ten sketches. How does Escher's choice of symmetry group change between the early (low-numbered) prints and the later (high-numbered) prints?
2. Explain why he made this deliberate change.
3. Do you find the patterns in the later sketches more satisfying?

13. Explain why Escher predominantly chose to use birds, fish, and lizards in his patterns.

14. Choose a sketch from Escher's regular division notebook (pages 116-229 of Visions of Symmetry or Regular Division of the Plane Drawings) that you particularly like. Explain what you like about it, and compare with similar sketches that don't work as well.

15. In Visions of Symmetry pg 31-36 there is a discussion of how Escher thought about tessellations, or as he called them: "regelmatige vlakverdeling" (regular divisions of the plane). Escher used three posters when lecturing about his work on these regular divisions of the plane.
1. The first poster shows examples of tessellations from various cultures. What cultures are represented?
2. The second poster represents the geometric tessellations that Escher used as a starting point to create his more intricate tessellations. What are the six "primitive" tessellations?
3. The third poster explains and demonstrates the geometric motions that preserve shape: translation, rotation and glide reflection. What are the symmetry groups of the 5 examples Escher gives?
4. Visions of Symmetry talks about the explanations of the 5 tessellations in this third poster. The tessellation with only translation is the easiest to understand. After reading the explanations, how would you rank the level of diffuculty of the other tessellations? Consider how hard it would be for you to explain the drawings to someone not in this class.

Instructor:Wallpaper Exercises Solutions (Instructors only).