Course:Outline and Study Guide for Exam - Bart07

The exam will consist of two parts. Part I (50% of the grade) will consist of about 4 problem taken from the list provided below. Part two will consist of another 5 problems where the problems (or parts of problems) will be taken from the homework.

Part I

I will ask some of the following questions. Remember that I am giving you exam questions, so I will not respond to questions like "What is the answer to problem 2?". You may of course ask more specific questions about the materials covered. Questions about the online text are appropriate.

Remember when preparing for the exam that this is a math class and even though we use art as a way to motivate the material, I will be looking for knowledge of mathematical concepts. When answering questions be complete and thorough. Draw pictures to illustrate what you are discussing if you think it may help get the point across. Make sure you use the correct mathematical terminology and vocabulary.

When studying you should use the text (ie the online Escher text) as your main resource and Visions of Symmetry as your secondary source.

1. Show carefully why there are only 3 regular tessellations (Show that the angle sum for the angles in an n-gon is (n-2)180 ; Then argue that the angles in a regular n-gon must measure (n-2)180 / n ; Show that the angle of an n-gon must divide 360, and show that there are exactly 3 - i.e show there cannot be any others.)
2. Carefully show that all triangles tessellate. Carefully go through as much detail as possible. Your answer should provide some detail of the arguments involved.
3. Carefully show that all quadrilaterals tessellate. Carefully go through as much detail as possible. Your answer should provide some detail of the arguments involved.
4. Describe the 4 isometries of the plane. Give examples. Describe how the isometries are used to create escher-like tessellations. Given a geometric tessellation (square or parallelogram) create an Escher-like tessellation. I will leave it up to you how (un)complicated to make this.
5. We discussed three different types of symmetry groups: rozette groups, frieze patterns (= border patterns) and wallpaper patterns. Define reflectional and rotational symmetry, and explain how they are used to deetermine the symmetry groups. Explain the similarities and differences between the different symmetry groups.
6. Given one or more of the tetrominoes find the rotational and reflectional symmtries it has. Using these tetrominoes create a tessellation and state what type of symmetries the tessellation has.
7. Describe Spherical geometry in your own words. Explain some of the differences with euclidean geometry. What are the differences between the axioms for Spherical and Euclidean Geometry? What is the sum of the angles in a triangle? What type of n-gons exist on the sphere? Are there n-gons that do not exist in Euclidean space? If so, what are they? Do we get all the same polygons? For instance, are there squares, rectangles etc? What are the isometries of the sphere? Discuss Escher’s art based on spherical geometry. Explain as much as you can, and provide illustrations if possible.

Part II

I will take some problems from homeworks 1, 2, 3, and 4.