Course:Math 124 F07 Bart Homework 4

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Read Visions of Symmetry pages 243 - 247: Regular division on three-dimensional surfaces.

Answer the following questions from Spherical Geometry Exercises: #2, 33, 37, 38, 39.

2. Why was Escher unhappy with tessellations of the plane as a means to display infinity? Look at Escher’s Sphere with Fish, Sphere with Angels and Devils, Sphere with Eight Grotesques, and Sphere with Reptiles (pages 244 and 245 Visions of Symmetry ). Do you think these balls would give a good sense of infinity? Why or why not?

33. Examine Concentric Rinds, and focus on the outer sphere. It’s tessellated by many copies of one triangle.

  1. What are the corner angles of this triangle?
  2. What is it’s defect?
  3. How many copies of the triangle cover the sphere?
  4. How many vertices, edges, and faces does this tessellation have?

37. Consider the spherical tessellations created by Escher on pages 244 and 245 of Visions of Symmetry. Describe the rotational symmetry present in each of the tessellations. Record the degree(s) of rotation and what the center of rotation looks like. For instance the "Heaven and Hell" sphere shows a 2-fold rotation about the bottom of the feet of the bats / angel.

Go to the Escher Gallery on the website maintained by Jos Leys [Escher Gallery]

38. Compare Escher's Clown Drawing to Leys' spherical tessellation. What rotational symmetries are present in the flat tessellation? What rotationa are present in the spherical tessellation [Clowns 04]?

39. Compare Escher's E15 (drawing with dragons) with the flat tessellation created by Leys [Dragons 01] and the spherical tessellation [Dragons 03]. What did Leys do to adapt the tessellation to the sphere?