Problem 1 One can define a line as the shortest distance between two points. On curved surfaces such lines are no longer straight when viewed from an extrinsic or external viewpoint (see Problem 2). Nevertheless, shortest distance lines do exist on curved surfaces. Using this definition, what is the line between Tallahassee, Florida and Multan, Pakistan on the surface of a perfectly round spherical globe?
Problem 2 In Euclid's Elements, a line is defined as having breadthless length while a straight line is defined as a line which lies evenly with the points on itself. A straight line on the surface of a sphere must curve from an extrinsic or external viewpoint, but intrisically, say for example if we are living in Kansas, we can define what it means to feel straight. What is straight on a sphere? Is the equator a straight line? Is the non-equator latitude between Tallahassee, Florida and Multan, Pakistan a straight line?
Problem 3 In Book 1 of Euclid's Elements, postulate 1 says that a straight line can be drawn from any point to any other point. Is this true on the sphere? Although it doesn't explicitly say so, since Euclid uses postulate 1 to say that there is a unique line between any two points, he really ought to have stated the uniqueness explicitly. Is it true that a unique straight line can be drawn between any two points on the surface of a sphere?
Problem 4 In Book 1 of Euclid's Elements, postulate 2 says that we can produce a finite straight line continuously in a straight line. (see also Problems 1 and 2). In modern language, we say that every straight line can be continued indefinitely. Is postulate 2 true on the surface of a sphere?
Problem 5 In Book 1 of Euclid's Elements, postulate 3 says that we can describe a circle with any center and radius. Is postulate 3 true on the surface of a sphere? In Book 1 of Euclid's Elements, postulate 4 says that all right angles equal one another. Is postulate 4 true on the surface of a sphere?
Problem 6 In Book 1 of Euclid's Elements, postulate 5 says that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Is postulate 5 true on the surface of a sphere?
Problem 7 In Book 1 of Euclid's Elements, proposition 31 says that given a straight line and a point off of that line, we can construct a straight line that is parallel to the given line and goes through the point. A corollary to proposition 31, also known as Playfair's axiom, says that only one such line can be drawn parallel to the given line. Is proposition 31 true on the surface of a sphere? Is Playfair's axiom true on the surface of a sphere?
Problem 8 In Book 1 of Euclid's Elements, proposition 4 (SAS or side-angle-side) says that if two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Is SAS always true for spherical triangles (a curved triangle formed by straight lines on the surface of the sphere)?
Problem 9 In Book 1 of Euclid's Elements, the second part of proposition 32 says that the sum of the three interior angles of the triangle equals two right angles. On the surface of a perfectly round beach ball, can the sum of the angles of a spherical triangle (a curved triangle formed by straight lines on the surface of the sphere) ever be greater than 180 degrees? Why?
Problem 10 In Book 1 of Euclid's Elements, proposition 47 says that in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Assume we have a right-angled spherical triangular plot of land (see Problem 9) 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?
Extra Credit Problem 1 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?
Extra Credit Problem 2 Is the surface of a sphere 2-dimensional or 3-dimensional? Why?
Extra Credit Problem 3 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?
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 WebCT posting and a presentation that will be due in 1 hour and 15 minutes. If you finish early, then work on the Extra Credit questions.
Your grade will be based on the quality of the references that you find, along with the clarity and depth of your answer in your presentation. Having the "right" answer is not of prime importance as it is often the case that at this stage of research, mathematicians will still have incorrect ideas. The idea here is to deeply explore your question with help from web searching, and then to clearly communicate your research.