Geometric Structures: Practice 7: Finite Projective Planes

prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

1. Consider the geometry consisting of triples (a, b, c) of elements from Z5, but (0, 0, 0) is not a point. These will be the points of a plane where points that differ by a multiple from Z5 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 Z5 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 lines x + 2z = 0 and 2x + 3y -z =0.

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

g. List all the points on the line at infinity for 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?

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