Research Problems

**Problem 2**
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 Tallahassee, Florida and Multan,
Pakistan an intrinsically straight path?

**Problem 3** 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 4
Besides 23 definitions and several implicit assumptions, Euclid derived much
of planar geometry from five postulates.
For thousands of years, people argued about the
necessity and validity of Euclid's 5th Postulate:
If a straight line crossing two straight lines makes the
interior angles on the same side less than two right angles, the
two straight lines, if extended indefinitely, meet
on that side on which are the angles less than the two right angles.
Is this true on the sphere? |

**Problem 6** 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 7**
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
Umanak, Greenland, Goiania, Brazil, and Harare, Zimbabwe, that
happens to measure 300 and 400 units on its short sides.
Is the measurement of the long side from Greenland to Zimbabwe
greater than, less than or equal to 500 units (ie is the
Pythagorean Theorem true on the sphere)? Why?

**Problem 8**
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 300 miles in each direction? What about 3000 miles in each direction?
Explain.

**Problem 9**
Is the surface of a sphere 2-dimensional or 3-dimensional? Why?

**Geometry of our Entire Universe**

**Problem 11** Is our universe 3-dimensional or is it
higher dimensional? Why?

**Problem 12** Are there are finitely or
infinitely many stars in the universe? Explain.

**Problem 13** The geometry that you explored in high school is
called Euclidean geometry. For example, you learned about
the Euclidean law stating that the shortest distance path
between two points is a straight line
and about the Euclidean law stating that
the sum of the angles in a triangle is 180 degrees.
Is our universe Euclidean
(ie does it satisfy the laws of Euclidean geometry such as
those just mentioned)?
How could we tell?

**Problem 14** While people thought that the earth was flat for a long
time, we know that the shape of the earth is actually a round sphere.
What is the shape of space (the universe)?

Project 2: Report DUE Tues at the beginning of class (NO lates allowed) Use your web searching skills to do internet and book research and/or physical experimentation to try and answer your question. I am happy to help you think of experiments and words to search for, but you should try and do so on your own first. You and your partner(s) should work together to prepare one report that includes:

Your Project 2 grade will be based on the quality of the
web and/or book references that you find and/or experiments that
you conduct, along with the clarity and depth of your answer.
**Having the "right" answer is not of prime importance** as
it is often the case that at this stage, mathematicians will still
have incorrect ideas. The idea here is to deeply explore your question with
help from web searching and/or experiments, and then to clearly
communicate your research.