# 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. Your answer should include:

- What are the differences between the axioms for Spherical and Euclidean Geometry? Be as detailed as you can.
- What is the sum of the angles in a triangle?
- What type of polygons exist on the sphere? Are there n-gons that do not exist in Euclidean space? Specifically discuss:
- Isosceles and equilateral triangles
- The square and rectangle. What definition are you using? What effect does that have?
- The rhombus and parallelogram. Do they exist? Why or why not?

- What do we know about regular and semiregular tessellations on the sphere? How many are there? How does that compare to Euclidean geometry?
- What are the isometries of the sphere? Draw an example.
- Discuss Escher’s art based on spherical geometry.

Explain as much as you can, and provide illustrations if possible. Think about what explorations introduced you to some of these concepts and what we discovered.

**2. Describe Hyperbolic geometry in your own words.** Explain some of the differences with euclidean geometry. Your answer should include:

- What are the differences between the axioms for Hyperbolic and Euclidean Geometry? Be as detailed as you can.
- What is the sum of the angles in a triangle?
- What type of polygons exist in hyperbolic space? Specifically discuss:
- Isosceles and equilateral triangles
- The square and rectangle. What definition are you using? What effect does that have?
- The rhombus and parallelogram. Do they exist? Why or why not?

- What do we know about regular and semiregular tessellations in hyperbolic space? How many are there? How does that compare to Euclidean geometry?
- What are the isometries of hyperbolic space? Draw an example.
- Discuss Escher’s art based on hyperbolic geometry.

Explain as much as you can, and provide illustrations if possible. Think about what explorations introduced you to some of these concepts and what we discovered.

**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.