# 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.