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** 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

- many different types of sources and diverse perspectives, including scholarly references and sources from the library and/or my office library
- 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 credibile the author is (empirical in presenting the thesis, good credentials, biased in any way)?

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.

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"