| 0. Email Introduction Due Monday, January 23 5 points Send me an email introducing yourself. Please give me some background on yourself, your academic experiences, and why you have enrolled in this course. I would like to get to know each of you and to help shape your experiences in this class to maximize your goals. My email address is oehrtman@math.asu.edu. |
| 1. What is Straight? Due Wednesday, January 25 10 points Read Chapter 1. 1. What are the symmetries of a straight line? For each one, what other shapes have this symmetry? 2. Can you find a small collection of symmetries of straight lines, for which ONLY straight lines have all of them? How can you convince someone of this? |
| 2. Straight on a Sphere? Due Wednesday, February 8 20 points Reread Chapter 2. 1. Pick three symmetries of a line on a plane and their analogs for lines in a sphere. Describe the isometries carefully for both the plane and sphere. That is, where does each point go? Identify all of the fixed points for each of these isometries in the plane and in the sphere. Explain why each isometry is a symmetry of a line in a plane and in a sphere. 2. Explain why a great circle is a geodesic on a sphere but a general latitude line is not. Give two different reasons. 3. Why does theorem 2.1 imply that all geodesics are great circles? 4. Visit the Foucault pendulum in PSF twice and record the location of the pendulum's swing. Use your data points to determine how much the pendulum rotates in exactly 24 hours. |
| 3. Angles Due Wednesday, February 15 20 points Read Chapter 3. 1. Prove the Vertical Angle Theorem (VAT): A pair of opposite angles formed by two intersecting straight lines are congruent. a. You need at least two different proofs for full credit. b. Each proof must state which definition of angle you are using. c. In addition, explain whether the proof will work on the sphere. I.e. if yes, explain if any modifications are necessary or why none are necessary. If no, explain why not. 2. Write three different definitions for angle: dynamic, measure, and geometric. For each definition a. State your definition completely. b. State what it means for the angles to be congruent using this definition. c. Discuss the applicability of the definition to the sphere (e.g. Does this definition work on the sphere? What modifications, if any, are necessary to make it work on the sphere?) |
| 4. Cones and Cylinders Due Wednesday, February 22 20 points Read Chapter 4. 1. Describe all of the isometries of a cylinder and a cone as functions. That is, your description should allow someone to determine where any point is mapped. What are the fixed points of each isometry? Are there any partial isometries (isometries from a portion of the space to itself)? Explain. 2. Describe all of the geodisics on cones and cylinders. 3. Can geodesics intersect themselves on cylinders and cones? How many times? Can there be more than one geodesic joining two points on cylinders and cones? How many different geodesics are there between any two given points? |
| 5. Intrinsic Derivative Due Wednesday, March 1 20 points 1a. Set up spherical coordinates for a sphere of radius r and parameterize a path for a circle at latitude φ0. 1b. Find the velocity and acceleration vectors for in 3-space for this path. Explain why the velocity vector is intrinsic but the acceleration vector is extrinsic. 1c. Find the covariant derivative by projecting the acceleration vector onto the tangent plane. 1d. Show that the covariant derivative is only zero at the poles and the equator. Explain what this means about geodesics on the sphere. 1e. The latitude of Tempe, AZ is 33°23' N. Determine the scalar speed (s) and the scalar intrinsic acceleration (a) for the path at this latitude. Recall that the rate at which v is turning is a/s. Determine this rate and multiply by the duration of one revolution to find the total amount rotation of the velocity vector. 2a. Consider the following coordinates for a cylinder of radius r x=rcosθ y=rsinθ z=h Describe what all paths look like on the cylinder that are parameterized by x=rcos(at) y=rsin(at) z=bt+c for different values of a, b, and c. 2b. Show that these paths are all geodesics on the cylinder by showing that the covariant derivative is zero. |
| 6. Congruence Theorems Due Wednesday, March 8 20 points Read Chapters 6 and 9. 1. Prove ITT and SAS for a plane. Afterwards, list all of the assumptions you needed to make. 2. Explain why SAS is not true for a sphere. Which assumptions from your list in Problem 1 do not hold for a sphere? Explain why restricting to "small triangles" resolves this problem. 3. Which of SSS, ASS, SAA, and AAA are true on the plane? Prove the ones that are true and give a counterexample to the ones that are false. |
| 7. Areas of Triangles Due Wednesday, March 29 20 points Read Chapter 7. 1. (Problem 7.1 in the book) a. The two sides of each interior angle of a triangle Δ 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, α, β, and γ, cover the sphere with some overlap. Describe what the overlap is. b. Find a formula for the area of a lune with lune angle θ in terms of θ and the (surface) area of the sphere (of radius ρ, which you can call Sρ. Use radian measure for angles. c. Find a formula for the area of a triangle on a sphere of radius ρ. 2. (Problem 7.3a in the book) What can you say about the sum of the interior angles of triangles on spheres? Are there maximum and/or minimum values for the sum? 3. Write up the proof that the holonomy of a small triangle on a sphere - and any triangle on the plane - is 2π minus the sum of the exterior angles (or the sum of the interior angles minus π). Follow the suggestions for problem 7.4 and the diagrams in Figure 7.9 in the text. |
| 8. Hyperbolic Triangles Due Wednesday, April 5 20 points Read Chapter 5. Re-read Chapter 7. 1. (7.2 in the book) a. Describe what you found when you drew the largest triangle you could on the hyperbolic plane. Would you have been able to make it include much more area if the ruffled part of the plane were extended significantly farther out? Explain. b. Show that on the same hyperbolic plane, all 2/3-ideal triangles with the same angle θ are congruent. c. Show that the area function Aρ is an additive function. That is Aρ(α+β)=Aρ(α)+Aρ(β) d. Find a formula for the area of a hyperbolic triangle. Explain your derivation thoroughly. You may assume the results from problem 16.4. e. Express the holonomy of a triangle on a hyperbolic plane as a function of the interior angles. Express the holonomy of a triangle on a hyperbolic plane as a function of the area of the triangle and the curvature -ρ-2. 2. (Problem 7.3a in the book) What can you say about the sum of the interior angles of triangles on hyperbolic planes? Are there maximum and/or minimum values for the sum? |
| 9. Euler Characteristic Due Wednesday, April 12 20 points Read all of Chapter 7, carefully reviewing any portions you have already read. Read pp. 228-229 from Chapter 17. 1. (From the discussion of Problem 7.2 in the book) Be able to explain in your own words the argument on page 84 that since Aρ is an additive function, then Aρ(α)=α*(Iρ /π) where Iρ is the area of an ideal triangle. You do not need to turn anything in for this problem. 2. (Problem 7.5 a-c in the book) Suppose that Γ is a simple polygon. Be able to explain in your own words why where K is the Gaussian curvature. You do not need to turn anything in for this problem. 3. (Problem 17.6 in the book) Prove that if a geometric manifold M has a geodesic cell-division then where the cell division has v vertices and e edges and f faces. Explain why the Euler number of any geodesic cell division of a sphere must be 2. 4. a. Draw a two-holed torus and mark locations with zero curvature, negative curvature, and positive curvature. Explain why you know those points have the designated curvature using the shape of the two-holed torus relative to the tangent planes at those points. b. Draw a cell division and compute the Euler Characteristic. c. Use the same cell division to compute the holonomy around each cone point on the cell division. Find the sum of the holonomies. d. Redraw the two-holed torus with a continuous vector field on it. Identify the zeros of the vector field. Draw each zero on a separate graph with a path around it showing how the vectors point at several locations along the path. Compute the index of each zero and find the sum of the indexes of all zeros. e. Explain how and why the results for parts b, c, and d are related. |
| 10. Higher Dimensions 4d_axes.doc (26.5 Kb) Due Wednesday, April 19 20 points Read Chapter 20. (Note this is chapter 22 if you are using the third edition.) 1. (Problem 20.1 in the book) How would you explain 3-space to a person living in two dimensions? 2. (Problem 20.2 b) Show that every great 2-sphere in the 3-sphere has reflection-in-itself symmetry. It will be easiest if you choose an orthonormal basis for R4 so that the great 2-sphere is in the 3-subspace spanned by the first three basis elements. You will need to draw several pictures and explain what isometry of the 3-sphere this particular symmetry corresponds to. It may be helpful to use axes as shown on page 271. A Word document with these axes may be downloaded from the link above. 3. (Problem 20.2 c) Show that every great circle has the symmetries in S3 of rotation through any angle and reflection through any great 2-sphere perpendicular to the great circle. This will be easiest if you choose an orthonormal basis for R4 so that the great circle is in the plane spanned by the first two basis elements. You will need to describe exactly what such a rotation does in each of the four projections. You will need to explain exactly what these rotations and reflections do. In order to do this, describe their effect on S3 using each of the 3-dimensional subspaces as shown on page 271. A Word document with these axes may be downloaded from the link above. 4. Explain how SO3, the set of rotations in 3-space, can be identified with the 3-dimensional unit ball with the antipodal boundary points identified. Describe several points in the ball and explain what rotations they correspond to. |
| 11. The Shape of Space Dodecahedron Net (27.5 Kb) dodecahedrons.doc (202 Kb) Due Wednesday, April 26 20 points + 10 extra credit points 1. (Problem 22.1 a-c in the Textbook) a. Could we show that the Universe is non-Euclidean by measuring the angles of a large triangle in our solar system? How accurately would we have to measure the angles? b. If the stars were distributed uniformly in space, how could you tell by looking at stars at different distances whether space was locally Euclidean, spherical, or hyperbolic? Explain. c. Suppose you know that certain types of stars (or galaxies) have a fixed known amount of brightness, and you can see several of these standard candles at various distances from Earth. How could you tell whether the Universe is Euclidean, spherical, or hyperbolic. Do not assume that the stars are uniformly distributed as in the previous problem. 2. (Problem 22.2 c and part of 22.5 b in the Textbook) Draw a picture similar to Figures 22.2 and 22.3, for the half turn manifold, which is the same as the quarter turn manifold except that it is obtained by gluing the top and bottom faces with a half turn. As in Figure 22.3, begin the geodesic in the middle of the bottom-right edge (A) and traveling to the center of the front face (B), labeling this segment 1. Continue the Geodesic until it starts repeating it's path, labeling each segment consecutively. Eplain how the pattern of circles of intersection for the sphere of last scatter would differ from this manifold and the three-torus. 3. (Modified from Problem 22.3 a-e in the Textbook) When you glue the opposite faces of a dodecahedron with a one-tenth clockwise rotation, how many edges are glued together? What if you use a three-tenths rotation? Or a one-half rotation? Download the Word document at the top of this assignment, and for each of the three gluings use a picture of a dodecahedron to i. label each vertex of the new manifold with letters (A, B, C,...) ii. label each edge with a different number of arrows. See Figure 22.5 for examples. The dihedral angle for a Euclidean dodecahedron is about 117°. Use this to explain which of the manifolds created by these three gluings are Euclidean, spherical, or hyperbolic. Then for each of the figures, show that the Euler Characteristic for the corresponding manifold is zero. 4. (Part of Problem 22.5 b in the Textbook) Consider the Poincare dodecahedral space - the geometric 3-manifolds created by the gluing from Problem 3 with ten sets of three edges being glued together at a dihedral angle of 120°. What would the pattern of matching circles look like if our physical Universe were the shape of this manifold and if the sphere of last scatter reaches far enough around the Universe for it to intersect itself? |