A. Questions for Submission

5.1 Euclidean and Non-Euclidean Geometry

State Euclid's 5th axiom as it was formulated by Playfair (5^ONE), as well as the two alternatives 5^NONE and 5^MORE.
Consider a geometry in which 5^ONE is replaced by 5^NONE. Prove that there exists a triangle in the latter geometry with its three angles summing to more than two right angles. (Hint: Draw a vertical line, and then extend two lines at right angles to it. What happens as you continue to extend those two lines?)
Two navigators meet at the equator and decide to try to prove that Earth is intrinsically curved. Explain how they can do this. Suppose that they can tell which direction is North, South East and West, and can in principle travel anywhere on the planet.

Is the surface of a cylinder extrinsically curved, intrinsically curved, or both? Explain.
What is geodesic deviation, and how does it allow us to identify positive, negative and zero curvature regions of a space?

Curvature. Identify which parts of the surface below have positive, negative and zero curvature.

Is there any sense in which the geometry and intrinsic curvature of space can be known independently of one's experience? Why or why not?
Geodesics. Why is a geodesic the non-Euclidean analogue of a straight line in Euclidean geometry?
Dropping the embedding space. Suppose we are considering a 4-dimensional spacetime in which there is no higher dimension for the space to curve into. Does it make sense to say that such a spacetime is curved?
Knowing geometry. Kant claimed that the axioms of Euclidean geometry constitute "synthetic a priori" knowledge. "Synthetic" means the knowledge does not just follow from the definition; "a priori" means the knowledge is accessible prior to experience. What does the discovery of non-Euclidean geometry imply about the existence of such knowledge? Would replacing Euclidean geometry with some more general non-Euclidean definition of geometry make "synthetic a priori knowledge" any more plausible?