Project 2: Connections and an Overview of Course Topics (Sphere Presentations) One Problem Per Group.

While geometry means measuring the earth, too often it is presented in an axiomatic way, divorced from reality and experiences. Over the course of the semester, we will have diverse and increasingly sophisticated assignments in order to satisfy the speaking designator on the class and to meet the catalog requirements of concept development and connections among mathemtaical perspectives. For this assignment, your grade will not based on the quality of your presentation. Instead it is based on your ability to reflect on what and how you presented.

In this project, we will use intuition from our previous geometry knowledge along with experiences from hands on models and searching skills. The purpose of this assignment is an introduction to the topics in the course syllabus as we explore diverse perspectives and connections and get to know each other:

A study of the development of Euclidean geometry through multiple perspectives, including synthetic and metric. Topics to be considered include parallelism, similarity, measurement, constructions, an axiomatic approach to polyhedra, and at least one non-Euclidean geometry. The course will focus on concept development and connections among mathematical perspectives. Prerequisites: MAT 1120 and either MAT 2110 or MAT 2510. (SPEAKING) Problem 1 A straight line on the surface of a sphere must curve from an extrinsic or external viewpoint, but intrinsically, say for example if we are living in Kansas, we can define what it means to feel like we are walking on a straight path. What is straight (intrinsically) on a sphere? Is the equator an intrinsically straight path? Is the non-equator latitude between Chicago and Rome an intrinsically straight path?

Problem 2 In Euclidean geometry, through a given point, only one line can be drawn parallel to a given line. Is this true on the sphere?

Problem 3 On the surface of a perfectly round beach ball, can the sum of angles of a spherical triangle ever be greater than 180 degrees? Why?

Problem 4 Assume that we have a right-angled spherical triangular plot of land (a curved triangle formed by three shortest distance paths on the surface of the sphere that also contains a 90 degree angle) on the surface of a spherical globe between approximately the north pole, a point on the equator, and a point one-quarter away around the equator. Do the sides satisfy the Pythagorean Theorem? Why?

Problem 5 On the surface of a perfectly round beach ball representing the earth, if we head 30 miles West, then 30 miles North, then 30 miles East, and then 30 miles South would we end up back where we started? Why? What about 3000 miles in each direction? Can we construct a square on a sphere? Explain.

Problem 6 Can we construct every convex polyhedron on a sphere, like a soccer ball [a spherical version of a truncated icosahedron]? Are there spherical polyhedra that have no flat equivalents?

Problem 7 Is SAS (side-angle-side) always true for spherical triangles on the surface of a perfectly round beach ball? Explain.

Problem 8 Similar triangles satisfy AAA (angle-angle-angle). Can you construct similar triangles on a sphere?

Problem 9 If we slice a perfectly round loaf of bread into equal width slices, where width is defined as usual using a straight edge or ruler, which piece has the most crust (or surface area)? Why?

Problem 10 Is the surface of a sphere 2-D or 3-D?

### Short Presentation - No More than 4 Minutes in Length

1. Introduce the group members.
2. Briefly review related content that applies to the geometry of the plane / Euclidean geometry / flat geometry.
3. Discuss why the topic is important in mathematics and the real-world.
4. Summarize diverse perspectives of your sphere problem in your own words.
I have texts in my office that may be helpful in addition to our course books and web searching you do. You will also have 4 minutes at the beginning of class on the Tuesday this is to to finalize or practice your presentation with your group members. I am always happy to help in office hours.

### Self-Reflection and Peer Review

Your grade will be based on the quality and depth of your formal and typed self-reflection which will be due the day after your presentation (typed bullet points are fine, but make sure to utilize formal writing):
1. Summarize the content of your portion of the presentation (if you created informal bullet points or notecards to use, these would suffice for this portion).
2. In terms of the clarity of the content you presented what aspects went well? What aspects could have been improved?
3. In terms of the depth of the content you presented, what aspects went well? What aspects could have been improved?
4. In terms of the creativity of your presentation style, what aspects went well? What aspects could have been improved?
5. In terms of the success of your presentation style, what aspects went well? What aspects could have been improved?
For each group that is not your own, write down (handwritten bullet points are fine):
• The names of the people in the group.
• At least two positive aspects of the presentation.
• At least one suggestion for improvement.

### Suggestions from Dr. Sarah

For part 2 of your presentation, you should briefly review related content that applies to Euclidean geometry (the geometry often explored in middle school and high school). For example, for Problem 1 you might review the definitions of lines and how to calculate them, for Problem 9 you might review some information related to area and surface area, for Problem 10, you might review coordinate systems... Both books and the web contain a lot of information on Euclidean geometry.

For part 3 of your presentation, searches like
"triangles are important"
or a search word related to your problem and then (with and without quotations) words like important, interesting, useful, or real-life applications. You might also pick some specific fields to see if there are applications:
Pythagorean theorem chemistry

For part 4 of your presentation, you do not need to prove an answer or completely resolve the issue on the sphere. You should look for various perspectives related to spherical geometry, and summarize those in your own words. Try different combinations of search terms related to your problem along with words like sphere, spherical, earth, or spherical geometry. Vary your word combinations:
Spherical Polyhedron
Polyhedra on a sphere
yield very different results, and quotations can be helpful if there are too many results:
"straight lines on a sphere"
In Wallace and West Roads to Geometry you can search in the appendix for double elliptic, sphere, and spherical geometry, which will all yield different pages related to some of these problems. In Sibley The Geometric Viewpoint see sections 1.6 and 3.5.