Mathematics 138
Problem Set 5 due Wednesday, March 16, 2005
1. For the small affine faculty senate, the proof in Ryan Appendix E is valid; so any collineation that fixes three points (instructors) is the identity and every collineation is an affine transformation.

a. Find the unique collineation that maps Cal into Eva, Gus into Bob, and Dan into Fay.  Make a table showing the image of each instructor and each committee.
b. Find a matrix A and vector b that represents this mapping in a coordinate system where Cal is (0,0), Gus is (1,0), and Dan is (0,1).  The entries in the matrix and vector will be integers modulo 3, so they will all be 0,1, or 2.  Hint: once you have the image of (1,0) and (0,1) you can write down the matrix!
2. For the medium affine senate, the proof in Ryan Appendix E is not valid, so there are collineations that hold three points (instructors) fixed but are not the identity.  Find such a collineation that holds Emma, Kate, and Phil fixed.  Make a table showing the image of each instructor and each committee.  Since the square of this collineation is the identity, it either holds a committee or instructor fixed or interchanges two committees or instructors; so your table only has to show the instructors and committees that get interchanged.
Hint: the solution is isomorphic to the one in the lecture notes.  Choose one instructor
(say Emma) as the additive identity for both axes, the other two as the multiplicative identities for the two axes respectively.
3. Prove that in an affine plane where each vector is its own additive inverse (such as the medium affine senate), the diagonals of any parallelogram are parallel.  Illustrate this theorem with an example from the medium affine senate.
4a. Ryan, problem 30 on page 68.  There are famous attempts to "prove" Euclid's parallel postulate that assumed this as an axiom, but it is equivalent to the parallel postulate.
b. Make a diagram to show that this theorem is not valid in hyperbolic geometry.  Use Model 6 from the Cayley-Klein incidence document, in which the points of hyperbolic geometry are the points inside the unit circle and the lines are Euclidean lines. On your diagram, indicate for what portion of the interior of the angle the theorem is valid. It is sufficient just to construct an angle from two rays, but once you see what is going on you can draw a diagram showing what happens for all four angles that are formed by two intersecting lines.
c. Make another diagram to show that this theorem is not valid in hyperbolic geometry.  This time, use Model 2 from the Cayley-Klein incidence document, in which the points of hyperbolic geometry are the points in the upper half plane and the lines are either vertical Euclidean lines or Euclidean semicircles centered on the x axis. On your diagram, indicate for what portion of the interior of the angle the theorem is valid.  It is sufficient just to construct an angle from two rays, but once you see what is going on you can draw a diagram showing what happens for all four angles that are formed by two intersecting lines.

5. Go to http://aleph0.clarku.edu/~djoyce/java/elements/elements.html for interactive access to Euclid's Elements.
a. Find the proposition in Euclid that is equivalent to Theorem 40 on page 65 of Ryan.  State Euclid's proof of it in your own words.
b. Trace Euclid's proof back to his postulates and thereby determine which of the five postulates this theorem relies on.  The best way to do this is by constructing a tree that shows, as "ancestors" of a proposition, the other propositions and postulates on which it depends.  Ignore the so-called "common notions." The hyperlinks on the Web site make this easy to do.

6. The complex numbers C form a field. Show that complex conjugation f(x + iy) = x - iy is an
automorphism of C. Which step in Ryan's proof from appendix E is not valid for the complex
numbers?
7a. Make a diagram showing how to bisect an angle using straightedge and compass, and prove that your construction is correct according to the definition of bisector on page 51 of Ryan.
b. Prove the correctness of the following technique for trisecting an angle, attributed to Archimedes.

The angle to be trisected is AOB.  Mark a segment on your straightedge whose length equals the radius r of the circle through A and B centered at O.  Slide the straightedge so that it passes through A and the marks are aligned with P (on line OB) and M on the circle. Then angle APB is one-third of AOB.
8. Ryan, problem 38 on page 69.  This will connect Yaglom' s treatment of Galilean geometry with Ryan's notation.  Yaglom uses shears with a vertical axis, however.
9. Yaglom, problem 2 on page 46.

10a. Yaglom, problem 3 on page 47.  This is rather vaguely worded.  Prove the following:
The "right triangle formulas" dAC = dAB cosg a,  dBC = dAB sing a
The addition formulas sing (a + b) = sing a cosg b + cosg a sing b and
		      cosg (a + b) = cosg a cosg b - 0 sing a sing b
The second term in the addition law for the cosine has a zero where the Euclidean formula
has -1 and the hyperbolic formula (Ryan p. 208) has +1.
b. Yaglom, problem 4 on page 53.  Ignore the vague last sentence; just prove the Law of Sines and the Law of Cosines.