**Geometric Structures (Spring 2006)**

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

email: malkevitch@york.cuny.edu

web page: www.york.cuny.edu/~malk

Finite Projective Plane

1. Consider the geometry consisting of triples (a, b, c) of elements from Z_{5}, but (0, 0, 0) is not a point. These will be the points of a plane where points that differ by a multiple from Z_{5 }are considered the same. The lines of the geometry have the form ax + by + cz = 0 where the coefficients a, b, and c are not all zero. Note, for simplicity, bars of the numbers in Z_{5} are omitted.

a. Find the equation of the line which joins the points (1, 2, 1) and (4, 1, 1).

b. Find the equation of the line which joins the points (2, 3, 0) and (1, 2, 1).

c. Find the equation of the line which joins the points (1, 1, 0) and (3, 4, 0).

d. Find the point of intersection of x + z = 0 and x + 2y + 3z = 0.

e. Find the point of intersection of the x + 2z = 0 and 2x + 3y -z =0.

f. (Challenge): Verify for a specific configuration that Desargues theorem holds in this plane.

2. Can you figure out how many points and lines there are in the plane above?

3. Can you figure out how many points and lines there are in a projective planes of order n, where there are n + 1 points on every line? How many lines where are when there are n +1 points on every line?