Problem 1 One can define a line as the shortest distance path between two points. On curved surfaces such paths are no longer straight when viewed from an extrinsic or external viewpoint (see Problem 2). Nevertheless, these paths do exist on curved surfaces. What is the shortest distance path (staying on the surface of the sphere) between Tallahassee, Florida and Multan, Pakistan on the surface of a perfectly round spherical globe?
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 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 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 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 300 miles in each direction? What about 3000 miles in each direction? Explain.
Problem 7 Is the surface of a sphere 2-dimensional or 3-dimensional? Why?Extra Credit If we slice one-half of a perfectly round loaf of bread 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 or is it higher dimensional? Why?
Problem 9 Are there are finitely or infinitely many stars in the universe? Explain.
Problem 10 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 11 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)?
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.
You and your partner should work together to prepare one report that is due at the beginning of class on Tuesday Nov 21. The report counts as 100% of this major writing assignment grade.
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.