Recognizable Tessellation Exercises

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  1. 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. Read Escher's essay "The Regular Division of the plane used in surface decoration" (Visions of Symmetry, page 77-78).
      Explain why he made this deliberate change in the symmetry he used.
    3. Do you find the patterns in the later sketches more satisfying?
  2. Read Escher's essay "The Regular Division of the plane used in surface decoration" (Visions of Symmetry, page 77-78).
    Explain why Escher predominantly chose to use birds, fish, and lizards in his patterns.
  3. 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.
  4. Describe what's going on in Verbum. How does Escher use black and white in this print? The seemingly different prints Verbum, Three Worlds, and Puddle all have a common theme. What is it?
  5. Find four Escher prints in which a black and white tessellation develops from a undefined grey or neutral field. What point is Escher making?
  6. 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.

  7. Escher-pegasus.gif

    The pegasus shown here is from Escher's Sketch #105 (Pegasus). It is based on a square (the corners of which are shown with circles). It tessellates using only translations.

    1. The pegasus' belly is also its ______________
    2. The front of its neck is also its ________________
    3. The space between its front legs also forms its _____________
    4. Its back legs fit between ________________________
    5. The ________________ fits into its open mouth.
  8. A tessellation by parallelograms is shown below. Two of the sides have been altered: one has become a circular arc, and one has become three line segments. Draw a tessellation with the parallelogram grid but use the new dark curves for all edges. Keep the vertical glide reflection symmetry of the original parallelogram pattern.

Instructor:Recognizable Tessellation Exercises Solutions