Project 1: Geometry of our Earth and Universe -
You may work alone or in a group of up to 3 people and turn in one project
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
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
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
Through a given point, only one line can be drawn parallel to a given line.
Is this true on the sphere?
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?
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.
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.
internet research, library and book research
and (if applicable) physical experimentation to try and answer
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
Here is a rubric for grading for this project.
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, diverse perspectives and annotations.
- many different types of sources containing
diverse and contradictory
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
- 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:
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:
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
Note that if you are on campus,
then you will have full access to the library's subscriptions from
The main library webpage is at
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.
Choose 2 of 3 Projects
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