# Exam 2 Outline

## 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. 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.

2. Describe Hyperbolic geometry in your own words. Explain some of the differences with euclidean geometry. What are the differences between the axioms for hyperbolic and Euclidean Geometry? What is the sum of the angles in a triangle? What type of n-gons exist in Hyperbolic space? 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 hyperbolic space? Discuss Escher’s art based on hyperbolic geometry. Explain as much as you can, and provide illustrations if possible.

3. What are the 5 axioms for each of our three geometries? Be able to give specific statements of the axioms. Explain what geodesics ("straight lines") look like as part of your explanation.

## Part II

Expect questions from the explorations and homework.