Review Examination II (Spring 2007)

Geometric Structures, Mathematics 244 (Part I)

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

1. Give examples of finite and infinite projective planes by describing the points and lines of such planes.

2. Give an example of a finite hyperbolic plane. How many points and lines does the plane have?

3. What are the three (incidence) axioms for:

a. An affine plane

b. A projective plane

c. A hyperbolic plane

4. a. Is it possible to define all the terms in a mathematical system?

b. What are the usual undefined terms for a geometry?

5. Draw a diagram of the Fano plane? How many points are there in this plane? How many lines? How many lines through each point?

6. Draw an affine plane with 4 points. How many lines does this plane have? How many lines are there through a point? If a parallel class in an affine plane consists of all the lines parallel to each other in the plane, list all the lines in each of the parallel classes of the plane that you draw.

7. Euclid's 5th postulate dealt with parallelism. Give a modern version of this axiom.

8. In Euclidean three space is it possible for lines in different planes not to have any point in common? (Lines of this kind are usually referred to as skewed lines.)

9. Evaluate the following determinants:

a.

and

b.

(Evaluate this in three ways: i. expand using the first row; ii. expand using the first column; iii. use column operations to get zeros in positions (1,2) and (1,3).)

c.

d.

10. In the real projective plane:

a. Find the Euclidean equivalent of these point if there is one:

i. (2, 3, -2)

ii. (4, 0 2)

iii. (-1, 2, 0)

iv. (0, 4, 2)

v. (0, 2, 4)

vi. (0, 4, 0)

b. Find the line through the given pair of points:

i. (2, 3, 4) and (-1, 2, 5)

ii. (-1, -1, -1) and (2, 3, -2)

iii. (0, 3, 0) and (1, 2, 3)

c. Find the point where the lines given meet:

i. 2x -y + 2z = 0 and -x + y - z = 0

ii. 3x + 2y - z = 0 and -x + y + 2z = 0

iii. y + z = 0 and -x + 2y - z = 0

iv. -x + 2z = 0 and x = 0

v. x = y and y = z

vi. 3x - y + 2z = 0 and 2x - y + z = 0

11. Draw a picture which illustrates Desargues Theorem in the real projective plane.

12. l: 2x -y + z = 0 and m: 2x -y -3z = 0 and n: 2x - y + 4z = 0 are all lines which go through a common point in the real projective plane.

a. Find the coordinates of this common point.

b. If triangle ABC is such that A = (1, 3,1) (on l) and B = (1, 5, -1) (on m) and (1, 6, 1) (on n) and A'B'C' is such that A' =(0, 1, 1) (on l), B' = (3, 0, 2) (on m) and C'= (1, 2, 0) (on n), verify that Desargues Theorem holds.

13. a. Find the value of a, if any, which will make the following lines concurrent:

x + y + z = 0, 2x + y - z = 0 and ax + 2y - z = 0.

b. Find the value of b if any which will make the following points collinear:

(2, 0, 3), (-1, 1, 2) and (0, b, -2).

14. Determine if the lines: x - y = 0, 3y + 2x - 5z = 0 and -x + 3y - 2 z = 0 are conncurrent.

15. Determine if the points (1, 2, 4), (2, 5, 1) and (3, 2, 1) are collinear.

16. Write down the equations of the three sides of the triangle X = (0, 1, 2), Y= (2, 4, 1) and z = (-1, 2, 1) in the real projective plane.

17. Is every projective plane Desarguian?

18. What lines in the real projective plane correspond to these Euclidean lines? What new point (no corresponding point in the Euclidean plane) does the line go through in the real projective plane?

a. 2x - y = 6

b. x = 3y -2

c. y = 6

d. x = 9

e. x + y - 4 = 0

19. a. Write down the multiplication and addition table for GF (5) = Z5.

Note: Below, all coefficients and numbers are in Z5.

b. Write down the points on the line x + y = 4 in the affine plane whose points are ordered pairs from Z5.

c. Write down a line parallel to the line above and goes through the point (1, 1).

d. What line corresponding to the line in b. which is in the projective plane with 31 points associated with the 25 point affine plane. (Hint: the points of this projective plane are ordered triples (x, y, z) taken from Z5.
e. What line passes through (1, 2, 1) and (2, 1, 2)?

f. At what point do the projective lines x + y + z = 0 and 2x + y + 3z = 0 meet?

g. List all the points on the line x + y + z = 0. (Hint: There are exactly 6 points, one with z = 0 and all the others with z = 1. Remember, for example that the point (2, 0, 3) is the same as the point (4, 0, 1).)

20. Find the roots if any, of the polynomial: x2 + x + 1 = 0. (Example: 2 is not a root because 2x2 + 2 + 1 is congruent to 2, not 0, modulo 5).

21. Since x2 + 4x + 1 = 0 has no root in GF (5), we can construct a finite arithmetic with 25 elements where the numbers have the form a + bÆ where Æ2 = -4Æ - 1 = 1Æ + 4. Remember that in GF (5) "-3" = "2" and "-2" = "3" and "-4" = "1."

In this field find:

a. The sum of 2 + 4Æ and 3 + 2Æ and the product of these two numbers. (Hint: Treat the terms like polynomials but use the fact that the coefficients are in GF (5) and also that Æ2 must be replaced by 4Æ + 1.)

b. Repeat for 1 + 4Æ and 4Æ.

22. Briefly discuss the difference between geometry thought of as a branch of mathematics and geometry thought of as a branch of physics?