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Station 1 (A)

Station 1 (A). Towards the beginning of the semester, many students made the following claim: “A straight line is the shortest distance between 2 points.” Explain why this definition fails to describe geodesics in general and why it is also insufficient on the plane. Station 1 (B).

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Station 1 (A)

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  1. Station 1 (A) Towards the beginning of the semester, many students made the following claim: “A straight line is the shortest distance between 2 points.” Explain why this definition fails to describe geodesics in general and why it is also insufficient on the plane.

  2. Station 1 (B) What can you say about the truth value of the biconditional statement below on the hyperbolic plane? Prove/disprove each direction. (Hint: One direction is always true and the other is always false.) Two geodesics l and l’ never intersect l and l’ are equidistant

  3. Station 2 (A) We say that the only geodesics on a cylinder are vertical generators, great circles and helixes. Prove that these are the only geodesics on a cylinder.

  4. Station 2 (B) Prove that PT! is false on a sphere.

  5. Station 3 (A) Describe Alexandria, its significance in ancient times, and its final demise.

  6. Station 3 (B) Of all the mathematicians studied in Math 381, who was the most influential and why? (Note: This is subjective. Be sure to justify your answer.)

  7. Station 3 (C) It is clear from the JTG reading that the author has a deep respect for Archimedes. Is he justified in glorifying Archimedes as he does? Be sure to give at least 3 reasons to back up your answer.

  8. Station 4 Prove or provide a counterexample to each statement: ITT holds true for all triangles having two legs of equal length on the sphere. ITT holds true for all triangles (with finite vertices) having two legs of equal length on the hyperbolic plane.

  9. Station 5 What two fundamental properties of the Euclidean plane (other than EFP) make it distinct from all the other spaces we studied? (Hint: One property relates to triangles, the other to non-intersecting lines.) Prove that if we assume Euclid’s 5th Postulate (EFP) to be true, then both the properties above hold. (That is, prove EFP implies both your answers above.)

  10. Station 6 • Given any two points (excluding the cone point) on a cone with angle less than , does there always exist at least one geodesic connecting them? Justify your answer with a proof.

  11. Station 7 • Given any two points (excluding the cone point) on a cone with angle greater than , does there always exist at least one geodesic connecting them? Justify your answer with a proof.

  12. Station 8 • Give a proof of ASA on the plane (using properties of geodesics and EG ideas – not Euclid’s Elements). • Does your proof also hold on the hyperbolic plane? Justify your answer using the properties of geodesics on the hyperbolic plane.

  13. Station 9 • Prove Proposition I.18 from Euclid’s Elements, using only his Definitions, Common Notions, Postulates, and the Propositions 1-17.

  14. Station 10 • State Playfair’s Postulate (PP). Hint: page 139 of EG. • For each space below, determine whether PP holds true. If it holds true, prove that it holds true. Be sure to state what you need to assume in order to prove it. If it does not hold true, explain why. • Euclidean plane • Sphere • Cylinder • Cone (with angle less than 2π), • Cone (with angle greater than 2π) • Hyperbolic space

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