Euclidean and Non Euclidean Geometry

Problem 5.1 What Is Straight in a Hyperbolic Plane?

a. On a hyperbolic plane, consider the curves that run radially across each annular strip. Argue that these curves are intrinsically straight. Also, show that any two of them are asymptotic, in the sense that they converge toward each other but do not intersect.

b. Find other geodesics on your physical hyperbolic surface. Use the properties of straightness (such as symmetries) you talked about in Problems 1.1, 2.1, and 4.1.

c. What properties do you notice for geodesics on a hyperbolic plane? How are they the same as geodesics on the plane or spheres, and how are they different from geodesics on the plane and spheres?

Hint for 5.1 a)

Some of you seem to be having trouble visualizing 5.1a. This is because on the hyperbolic soccer ball model that most of you made, you can't see the annuli. You can see the annuli on a crocheted hyperbolic plane, but so far only one person has turned in a crocheted plane. This YouTube video records the construction of an annular hyperbolic plane, which should help you visualize the asymptotic lines on the hyperbolic plane.

https://www.youtube.com/watch?v=yc2huxWQpdQ

Problem 7.1 and Problem 8.2 Due date: Feb 28th

7.1

The Area of a Triangle on a Sphere

a. The two sides of each interior angle of a triangle A on a sphere determine two congruent lunes with lune angle the same as the interior angle. 'Show how the three pairs of lunes determined by the three interior angles, a, f, y, cover the sphere with some overlap. (What is the overlap?)

Draw this on a physical sphere, as in Figure 7.2

b. Find a formula for the area of a lune with lune angle 6 in terms of 6 and the (surface) area of the sphere (of radius p), which you can call Sp. Use radian measure for angles.

Hint: What if <9is n? ji/2?

c. Find a formula for the area of a triangle on a sphere of radius p.

8.2

Problem 8.2 Symmetries of Parallel Transported Lines

Consider two lines, r and r', that are parallel transports of each other along a third line, l. Consider now the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure.

What can you say about the lines r and rl Do they intersect? If so, where? Look at the plane, spheres, and hyperbolic planes.

If a transversal cuts two lines at congruent angles, are the lines, in fact parallel in the sense of not intersecting?

Suggestions can be found on the book.

Problem 9.1, 9.2 and 10.1 Due date is March 6th

Problem 9.1 Side-Side-Side (SSS)

Are two triangles congruent if the two triangles have congruent corresponding sides?

Suggestions

Start investigating SSS by making two triangles coincide as much as possible, and see what happens. For example, in Figure 9.2, if we line up one pair of corresponding sides of the triangles, we have two different orientations for the other pairs of sides as depicted in Figure 9.2. Of course, it is up to you to determine if each of these orientations is actually possible, and to prove or disprove SSS. Again, symmetry can be very useful here. 6n a sphere, SSS doesn’t work for all triangles. The counterexample in Figure 9.3 shows that no matter how small the sides of the triangle are, SSS does not hold because the three sides always determine two different triangles on a sphere. Thus, it is necessary to restrict the size of more than just the sides in order for SSS to hold on a sphere. Whatever argument you used for the plane should work for suitably defined small triangles on the sphere and all triangles on a hyperbolic plane. Make sure you see what it is in your argument that doesn’t work for large triangles on a sphere.

Problem 9.2

Problem 9.2 Angle-Side-Side (ASS)

a. Are two triangles congruent if an angle, an adjacent side, and the opposite side of one triangle are congruent to an angle, an , adjacent side, and the opposite side of the other! Look at plane, spheres, and hyperbolic planes.

b. Show that ASS holds for right triangles on the plane {where the Angle in Angle-Side-Side is right).

This result is often called the Right-Leg-Hypotenuse Theorem (RLH), which can be expressed in the following way:

RLH: On the plane, if the leg and hypotenuse of one right

triangle are congruent to the leg and hypotenuse of another

right triangle, then the triangles are congruent.

What happens on a sphere and a hyperbolic plane?

Suggestions

Suppose you have two triangles with the above congruencies. We will call them ASS triangles. We would like to see if, in fact, the triangles are congruent. We can line up the angle and the first side, and we know the length of the second side (BC or B'Cj, but we don’t know where the second and third sides will meet.Here, the circle that has as its radius the second side of the triangle intersects the ray that goes from A along the angle a to B twice. So ASS doesn’t work for all triangles on the plane or spheres or hyperbolic plane6. Try this for yourself on these surfaces to see what happens. Can you make ASS work for an appropriately restricted class of triangles? On a sphere, also look at triangles with multiple right angles, and, again, define “small triangles” as necessary. Your definition of “small triangle”

Problem 10.1

Problem 10.1 Parallel Transport on the Plane

Show that if l\ and h are lines on the plane such that they are parallel transports along a transversal l, then they are parallel transports along any transversal. Prove this using any assumptions you find necessary. Make as few assumptions as you can, and make them as simple as possible. Be sure to state your assumptions clearly.

a. What part of your proof does not work on a sphere or on a hyperbolic plane?

Suggestions

This problem is by no means as trivial as it at first may appear. In order to prove this theorem, you will have to assume something — there are many possible assumptions* so use your imagination. But at the same time, try not to assume any more than is necessary. If you are having trouble deciding what to assume, try to solve the problem in a way that seems natural to you and then see what develops while making explicit any assumptions you are using.

On spheres and hyperbolic planes, try the same construction and proof you used for the plane. What happens? You should find that your proof does not work on these surfaces. So what is it about your proof (on a sphere and hyperbolic plane) that creates difficulties?

Problem 10.1 emphasizes the differences between parallelism on the plane and parallelism on spheres and hyperbolic planes. On the plane, non-intersecting lines exist, and one can “parallel transport” everywhere. Yet, as was seen in Problems 8.2 and 8.3, on spheres and hyperbolic planes two lines are cut at congruent angles if and only if the transversal line goes through the center of symmetry formed by the two lines. That is, on spheres and hyperbolic planes two lines are parallel transports only when they can be parallel transported through the center of symmetry formed by them. Be sure to draw a picture locating the center of symmetry and the transversal. On spheres and hyperbolic planes it is impossible to slide the transversal along two parallel transported lines while keeping both angles constant (something you can do on the plane). In Figures 10.1, the line r' is a parallel transport of line r along line /, but it is not a parallel transport of r along /'.

We will now divide parallel postulates into three groups: those involving mostly parallel transport, those involving mostly equidistance, and those involving mostly intersecting or non-intersecting lines. This division is useful even though it is rough and unlikely to fit every conceivable parallel postulate.

b. Which of these three groups is the most appropriate for your assumption from Problem 10.1? We will call the assumption you made in Problem 10.1 “your parallel postulate.’’’’