# Project 1: Geometry of our Earth and Universe - Annotated Bibliography

You may work alone or in a group of up to 3 people and turn in one project per group.

While geometry means measuring the earth, too often it is presented in an axiomatic way, divorced from reality and experiences. In this segment we will use intuition from experiences with hands on models and we will develop our web searching research skills in order to understand real-world applications of geometry such as the geometry of the earth and universe and applications of geometry to art. You are going to do some research in mathematics the way that mathematicians do. We first think about the problems by ourselves. Then we consult books and journals, and rethink the problem using ideas from other sources to help us. Eventually we might talk to an expert in the field and see if they have ideas to help us. This process can be frustrating, but that it is the struggle and the process itself that leads to true understanding.

Research Problems - Choose One Problem to Research

Geometry of our Earth

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 For thousands of years, people argued about the necessity and validity of Euclid's Parallel Postulate. One form of this postulate is given as Playfair's Axiom: 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 (a curved triangle formed by three shortest distance paths on the surface of the sphere) ever be greater than 180 degrees? Why?

Problem 4 Is SAS (side-angle-side, which says that if 2 sides of a triangle and the angle between them are congruent to those in a corresponding triangle, then the 2 triangles must be congruent) always true for spherical triangles (a curved triangle formed by three shortest distance paths) on the surface of a perfectly round beach ball? Explain.

Problem 5 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 6 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 make a square on a sphere? Explain.

Geometry of our Entire Universe

Problem 8 Is our universe 3-dimensional or is it higher dimensional? Why?

Problem 9 Are there are finitely or infinitely many stars in the universe? Explain.

Problem 10 We know that the shape of the earth is close to a round sphere. Could the universe be round too? Does it have any kind of shape?

Project 1: Annotated Bibliography DUE at the beginning of class (NO lates allowed) Choose one problem. You may work alone or in a group of up to 3 people. Conduct internet research, library and book research and (if applicable) physical experimentation to try and answer your question. I am happy to help you think of experiments and help you find references, but you should try and do so on your own first.

Create an annotated bibliography with the annotations in your group members' own words providing

1. many different types of sources and diverse perspectives, including scholarly references and sources from the library and/or my office library
2. annotations that explain how the material in the source relates to your question.
3. an evaluation of the source, including how current it is and how credibile the author is (empirical in presenting the thesis, good credentials, biased in any way)?
Having the "right" answer is not of prime importance as it is often the case that at this stage, mathematicians will still have incorrect or contradicting ideas. The idea here is to deeply explore your question with and then to clearly communicate the diverse perspectives you found.

The bibliography and annotations must be in a scholarly, professional and consistent format and style of writing, and you will be graded on the depth and clarity of your research and annotations.

### Research Suggestions

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, spherical geometry, or double elliptic 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"

• I have many helpful books in my office - stop by during office hours.
• In addition to the usual web engine searches, from the Advanced Search on Google, you can search in Google Scholar. Note that if you are on campus, then you will have full access to the library's subscriptions from Google Scholar
• The main library webpage is at http://www.library.appstate.edu/. You can click on Catalog and search there.
• From the main library webpage, you can click on Databases & Articles then on J and then search JSTOR. If you are off campus, then you will need to enter your banner id.
• The library database CQ Researcher has a Pro/Con for select topics and questions.