Mathematics 138
Problems due Wednesday, April 27, 2005

You can use SmProj.exe and LgProj.exe, Windows programs downloadable from the Web site, to check some of your answers. Luke Gustafson's PHP software, accessible via "PHP Affine," does
many of the same things  directly in your browser and is more portable.
1. For the "medium projective senate" with the coordinates given in the notes, use cross products to answer the following questions. The answers can of course be checked by using the committee listings.  In calculating, remember that u2 = u+1, (u+1)2 = u, and u(u+1) = 1.
a. What is the equation of the committee s on which Carl and Lynn serve? Check that Fred and Seth both satisfy this equation.
b. What is the equation of the committee t on which Beth and Irma serve?
c. What is the equation of the committee u on which Dave and Neil serve?
d. What instructor serves on both s and t?  On both s and u?
2. Suppose that the proof of Desargues' Theorem from the notes is applied to the large projective senate, with p = 1 , q = 2, r = 3.  All arithmetic is in Z5, of course.
a. Who are P', Q', and R'?
b. Who are L, M, and N, and on what committee do they all serve?
3. a. State the converse of the projective version of Desargues' theorem, and show that it is the same as the dual of Desargues' theorem.
Either b1. If you have Mathematica, do a direct brute-force proof of the converse of Desargues' theorem like the one attached to the lecture notes, and hand in a printout.
Or b2. Prove the converse of Desargues' theorem directly from Desargues' theorem, without using coordinates (this is the right way to establish that the projective axioms are self-dual)

4. State and prove the projective dual of Pappus' theorem. Draw a diagram to illustrate the theorem (you'll probably want a separate sheet of paper for this, and it may still take some trial and error to get all the lines to intersect on the page.)  It's fine to get assistance from Mathematica, in which case you can use brute force.  If you do a coordinate proof by hand, use the opportunity to assign simple coordinates to four points. For a real challenge, try to do a proof directly from the projective axioms.  This is hard, because Pappus' Theorem itself is then an axiom and you don't have a proof of it to dualize! (There may be a good term project here -- what is the converse of the dual?)

5a. Draw a diagram to illustrate the projective version of Pappus' Theorem for the case where the
line A'B'C' is chosen as the line at infinity.
b. Draw a diagram to illustrate the projective version of Pappus' Theorem for the case where the
line AB'P is chosen as the line at infinity.
6a. Ryan, problem 7 on page 138. This is the classic approach to projective collineations, closely related to the discovery of perspective by Renaissance artists.
b. For the large projective senate, with P = Danny, Q = Maria, P' = Chris, Q' = Quint, describe the "unique perspectivity" of part v of the problem of part a, and find the image of Terry.

7. For the large projective senate, find a matrix A that maps Zelda to Viola, 5larm to Kevin, Alice to
Oscar, and Betty to Chris.  There are four possible answers: choose the one that has an entry of 1
in the upper left-hand corner.
This is harder to do than you might have guessed, since expressing a vector as a linear combination
of three basis vectors is equivalent to inverting a 3x3 matrix. In this case there are enough zeroes so that the computation is easy. Check your answer with LgProj.exe or with the PHP software.
8. A polarity of the small projective senate has the following lightlike instructors and committees:
Eva on the library committee.
Hal on the junior committee.
Jim on the athletics committee.
Ken on the housing committee. 
a. Which are the six spacelike committees, and what instructor is the pole of each of them?
b. Who are the three timelike instructors, and what committee is the pole of each of them?
c. Find an orthogonal basis: a set of three instructors, two spacelike and one timelike, with the
property that each is the pole of the committee on which the other two serve.
d. What matrix B leads to this polarity?  Hint: find the images of (1,0,0), (0,1,0), and (0,0,1), then adjust the columns so that the matrix is symmetric.  Once you have B, you can use SmProj.exe or the PHP software to confirm all your answers.

9. For the polarity of the large projective senate defined by the matrix B =  
a. Use LgProj.exe to find the lightlike committees and instructors.
b. Construct the pole of the spacelike education committee using just incidence properties. (You
can check your answer using LgProj.exe, and you can use it to calculate intersections of committees and committees on which given instructors serve.  Draw a diagram to illustrate your construction.
c. Similarly, construct the polar line of the spacelike instructor Sally.

10a. Construct the inverse of the matrix B from the preceding problem.
b. Construct the polar line of the timelike instructor Larry using the matrix B. Then choose three spacelike committees on which Larry serves, and confirm that their poles are all on the polar line of Larry. Draw a diagram to illustrate the situation.
c. Construct the pole of the timelike library committee using the matrix B-1. Then choose three spacelike instructors on the library committee, and confirm that their polar lines all include the pole that you found.