Geometry of our Earth and Universe Questions

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 about the Earth and a Second Problem about the Universe

Geometry of our Earth (Choose one)

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 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?

Geometry of our Entire Universe (Choose one)

Problem 5 Is our universe 3-dimensional or is it higher dimensional? Why?

Problem 6 Are there are finitely or infinitely many stars in the universe? Explain.

Problem 7 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?

Assignment and Grading

Conduct scholarly research to look for at least two different, if possible contradictory, perspectives on the earth question, and at least two perspectives on the universe question. Keep track of your sources. Summarize what you found in the ASULearn assignment. I'll grade this on a Padawan (still training), Jedi, or Jedi Master scale using the criteria of scholarly sources, diverse perspectives, and research on both questions. This counts as a part of your homework grade.

Research Suggestions

You should look for various perspectives related to your research question, and summarize those in your own words. Try different combinations of search terms along with words like universe, space, 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" or "straight on a sphere"
  • 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 Books and Media and search there.
  • From the main library webpage, you can click on Databases & E-Research Tools 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.