### Dr. Sarah's Geometry of our Earth

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 an intrinsically 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 intrinsically 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?

Problem 6 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 7 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 8 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 9 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 intrinsically straight lines on the surface of the sphere)? Why?

Problem 10 In Book 1 of Euclid's Elements, proposition 8 (SSS or side-angle-side) says that if two triangles have two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Is SSS always true for spherical triangles (a curved triangle formed by intrinsically straight lines on the surface of the sphere)? Why?

Problem 11 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 intrinsically straight lines on the surface of the sphere) ever be greater than 180 degrees? Why?

Problem 12 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? Why?

Problem 13 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? Why?

Problem 14 Is the surface of a sphere 2-dimensional or 3-dimensional? Why?

Problem 15 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?

Using web and book references along with experimentation, research the answers to your questions. Prepare to present your research to the rest of the class, as part of the speaking designator and as a project grade. Create notes (informal notes on index cards are fine) for your presentation, which you will turn in afterwards. A good presentation should 1) Discuss the problem in your own words.
2) Summarize your research efforts. What web searches did you perform (what words did you use, and which combinations of words were or were not successful)? How did you find relevant books?
3) Explain what you found. Clarity and depth are important. If you found conflicting answers to your problem, summarize the various viewpoints in your own words.
4) Conclude by briefly summarizing the problem statement and the answer(s).
Adapted from Ian Parberry's Speaker's Guide: Oral presentations my be summed up as follows: "Tell them what you're going to tell them. Tell them. Then tell them what you told them". In the Introduction you tell them what you are going to tell them. In the Body and Technicalities you tell them. In the Conclusion you tell them what you told them. Don't be scared of this repetition. Sometimes repetition is the only way to clarify misconceptions. Naturally, this means that you should repeat things in different ways, and not quote yourself verbatim.

The Introduction This is possibly the most important part of your presentation. It sets the tone for the entire talk. It determines whether the audience will prick up their ears, or remain slumped in their chairs. A lot of snap decisions about your competency are made before the Introduction is over. First impressions are very important.
Define the Problem An amazing number of speakers forget this simple point. If the audience doesn't understand the problem being attacked, then they won't understand the rest of your talk.
Motivate the Audience Explain why the problem is so important. Throw in a little philosophy if necessary. How does the problem fit into the larger picture? What makes the problem interesting? You can return to these issues in the Conclusion, when you can re-address them with the benefit of hindsight.
Provide a Road-map Give the audience a brief guide to the rest of the talk

Background Material You should explain any background material that is necessary. You should also summarize important ideas from class that are necessary.

Main Material As you are writing, you need to say EVERYTHING you are writing, and pause at times to make eye contact. Orally reiterate parts and explain where you are going next. Be sure to go very slowly since others will not have seen your approach before.
Maintain Eye Contact Maintain eye contact with your audience. Spread your attention throughout the audience instead of concentrating on any one person or group (even if they are the only ones who matter). A good strategy for beginners is to choose a few people at random in different places in the audience, and look at them successively.
Control Your Voice Speak clearly and with sufficient volume. Don't speak in a monotone voice. Avoid information-free utterances ("Um, ah, er", etc.) Avoid fashionable turns of phrase. Avoid hype.
Control Your Motion Project energy and vitality without appearing hyperactive. Use natural gestures. Try not to remain rooted in one spot, but avoid excessive roaming. Don't stand in front of what you are writing.

The Conclusion Your aim here is to round off the talk neatly. You should discuss the results briefly in retrospect placing emphasis where it is needed. Hindsight is Clearer than Foresight. You can now make observations that would have been confusing if they were introduced earlier.

Indicate that your Talk is Over An acceptable way to do this is to say "Thank-you. Are there any questions?".