Station 1 (A)Station 1 (A)

Towards the beginning of the semester, Towards the beginning of the semester, many students made the following claim:many students made the following claim:

““A straight line is the shortest distance between 2 points.”A straight line is the shortest distance between 2 points.”

Explain why this definition fails to describe Explain why this definition fails to describe geodesics in general and why it is also geodesics in general and why it is also insufficient on the plane.insufficient on the plane.

Station 1 (B)Station 1 (B)

What can you say about the truth value of the What can you say about the truth value of the biconditional statement below on the biconditional statement below on the hyperbolic plane? Prove/disprove each hyperbolic plane? Prove/disprove each direction. direction. (Hint: One direction is always true (Hint: One direction is always true and the other is always false.)and the other is always false.)

Two geodesics Two geodesics ll and and l’l’ never intersect never intersect ll and and l’l’ are equidistant are equidistant

Station 2 (A)Station 2 (A)

We say that the only geodesics on a cylinder We say that the only geodesics on a cylinder are are vertical generatorsvertical generators, , great circlesgreat circles and and helixeshelixes. Prove that these are the . Prove that these are the onlyonly geodesics on a cylinder. geodesics on a cylinder.

Station 2 (B)Station 2 (B)

Prove that Prove that PT!PT! is false on a sphere. is false on a sphere.

Station 3 (A)Station 3 (A)

Describe Alexandria, its significance in Describe Alexandria, its significance in ancient times, and its final demise.ancient times, and its final demise.

Station 3 (B)Station 3 (B)

Of all the mathematicians studied in Math Of all the mathematicians studied in Math 381, who was the most influential and 381, who was the most influential and why? why?

(Note: This is subjective. Be sure to justify your answer.)(Note: This is subjective. Be sure to justify your answer.)

Station 3 (C)Station 3 (C)

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

Station 4Station 4

Prove or provide a counterexample to each Prove or provide a counterexample to each statement:statement:

ITT holds true for all triangles having two legs of ITT holds true for all triangles having two legs of equal length on the sphere.equal length on the sphere.

ITT holds true for all triangles (with finite vertices) ITT holds true for all triangles (with finite vertices) having two legs of equal length on the having two legs of equal length on the

hyperbolic plane.hyperbolic plane.

Station 5Station 5

What two fundamental properties of the Euclidean What two fundamental properties of the Euclidean plane (other than EFP) make it distinct from all plane (other than EFP) make it distinct from all the other spaces we studied? the other spaces we studied? (Hint: One (Hint: One property relates to triangles, the other to non-property relates to triangles, the other to non-intersecting lines.)intersecting lines.)

Prove that if we assume Euclid’s 5Prove that if we assume Euclid’s 5 thth Postulate Postulate (EFP) to be true, then both the properties above (EFP) to be true, then both the properties above hold. (That is, prove EFP implies both your hold. (That is, prove EFP implies both your answers above.) answers above.)

Station 6Station 6

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

2

Station 7Station 7

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

2

Station 8Station 8

Give a proof of ASA on the plane (using Give a proof of ASA on the plane (using properties of geodesics and EG ideas – not properties of geodesics and EG ideas – not Euclid’s Elements). Euclid’s Elements).

Does your proof also hold on the hyperbolic Does your proof also hold on the hyperbolic plane? Justify your answer using the properties plane? Justify your answer using the properties of geodesics on the hyperbolic plane.of geodesics on the hyperbolic plane.

Station 9Station 9

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

Station 10Station 10 State Playfair’s Postulate (PP). State Playfair’s Postulate (PP). Hint: page 139 of Hint: page 139 of

EG.EG. For each space below, determine whether PP For each space below, determine whether PP

holds true. holds true. If it holds trueIf it holds true, prove that it holds , prove that it holds true. Be sure to state what you need to assume true. Be sure to state what you need to assume in order to prove it. in order to prove it. If it does not hold trueIf it does not hold true, , explain why. explain why.

Euclidean planeEuclidean plane SphereSphere CylinderCylinder Cone (with angle less than 2Cone (with angle less than 2ππ),), Cone (with angle greater than 2Cone (with angle greater than 2ππ)) Hyperbolic spaceHyperbolic space