Problem 1 In Euclidean geometry, a line can be defined as the shortest distance between two points. On other spaces, one can also define a "line" as the shortest distance between two points. Using this definition, what is the "line" between Tallahassee, Florida and Multan, Pakistan on the surface of a round spherical globe (no bumps)? What do "lines" between other points look like on the surface? Explain.
Problem 2 On the surface of a perfectly round beach ball, can the sum of angles of a spherical triangle (a curved triangle on the surface of the sphere) ever be greater than 180 degrees? Why?
Problem 3 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 (see Problem 3) on the surface of a perfectly round beach ball? Explain.
Problem 4 Assume we have a right-angled spherical triangular plot of land (see Problem 3) on the surface of a spherical globe between approximately Umanak, Greenland, Goiania, Brazil, and Harare, Zimbabwe, that measures 300 and 400 on its short sides. How long is the long side from Greenland to Zimbabwe? Why?
Problem 5 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 6 Is the surface of a sphere 2-dimensional or 3-dimensional? Why?
Problem 7 If we slice one-fourth of a perfectly round loaf of bread (ie one quadrant) into equal width slices, where width is defined as usual using a straight edge or ruler, which piece has the most crust? Why?
Geometry of the Entire Universe
Problem 8 Is our universe 3-dimensional? Why?
Problem 9 How could we tell whether there are finitely or infinitely many stars in the universe? Explain.
Problem 10 Does our universe satisfy the laws of Euclidean geometry? Why?
Problem 11 How could we tell what shape space (the universe) is? Explain.
Problem 0:
How could we tell that the earth is round and not flat without using
any technology (ie if we were ancient Greeks)?
For my problem, I am asked how we could know that the earth is round
and not flat
without using any technology. I will attempt to answer this question
by using only my initial intuition.
As I first thought about this problem, it occurred to me that
if we traveled around the earth and
fell off of it while we were traveling, then we would know that
the earth was not round.
On the other hand, if we never fell off while
traveling, then we could
not tell whether the earth was round, flat
or some other shape.
It could still be flat but perhaps our travels had
just not taken us to the edge.
Historically, I think that people thought that the earth was indeed
flat, and that a ship could fall off the edge.
I then realized that this approach would not
solve the question, because it would never allow us to
determine that the earth is actually round.
I next thought about trying to find a definitive method to tell if the earth was round and not flat. If we could travel all the way around the earth, being assured that we were traveling in the same direction all the time, then this would differentiate our living on a round earth from living on a flat earth. Yet, we are not allowed to use any technology to help us, so a compass would not be allowed. Given this, I'm not sure how we could know that we were traveling in the same direction. Hence, I decided that while this was a good idea, I could not make the method work without technology.
Finally, I gave up on the idea of traveling to reach a specific destination, and started to think about the constellations. If we travel to different places on the earth, we see differences in the stars. For example, constellations look very different in the northern hemisphere than in the southern hemisphere. Also, even within the northern hemisphere, the north star is in different positions in the sky. This would not occur if the earth were flat and would indicate that the earth was round.
This concludes my initial intuition on Problem 0.
Prepare a report and presentation due at the start of class on Tuesday, November 13. The report counts as 100% of this major writing assignment grade. You will turn in two copies of the following: (1 for me to keep)
Your major writing assignment 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 - recall from the Fermat video that Shimura made "good mistakes". The idea here is to deeply explore your question with help from web searching and/or experiments, and then to clearly communicate your research. Be sure to follow the writing checklist guidelines.