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

**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**

**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?

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

- many different types of sources containing
**diverse and contradictory perspectives, including scholarly and nonscholarly references and books**and sources from the library and/or my office library [******specify which ones are from a library**in your annotations****] - annotations that explain how the material in the source relates to your question.
- an evaluation of the source, including how current it is and how credible the author is (empirical in presenting the thesis, good credentials, biased in any way)?

Here is a sample bibliographic annotation related to the topic of squares on a sphere:

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"

You will choose two projects (of the three options) to write up carefully by their respective due dates. You may complete more for extra participation credit, and your two highest grades will count. Research has shown that projects are extremely beneficial in learning and applying academic knowledge, so I'll ask everyone to do some related activities for homework, regardless of whether you choose to complete that project.

The projects are those listed on the calendar page, and I encourage you to complete the ones you find most interesting!

Project 1: Earth and Universe - Annotated Bibliography

Project 2: Benjamin Franklin's Financial Legacy

Project 3: Critical Analysis of Recent Media