One of the salient features of projective planes is that there is no possibility for parallelism, since any two lines must intersect.
PROJECTIVE PLANES
1. Two distinct points are contained in a unique line.
2. Two distinct lines intersect a unique point.
3. There exist four points of which no three are incident with the same line.
|
Is the same true for projective spaces?
PROJECTIVE SPACES
1. Two distinct points are contained in a unique line.
2. A line that intersects two sides of a triangle but does not contain any vertices of the triangle also intersects the third side of the triangle.
3. Every line contains at least three points.
|
That is, must any two lines of a projective space intersect? We will see later that any two such lines must lie in a projective plane that is a subspace of the projective space, and so must intersect. That is, the closure of any two lines is a projective plane (of dimension two, of course).
By removing a point and all the lines that are incident with it from a projective plane, we arrive at an affine plane:
AFFINE PLANES
1. Two distinct points are contained in a unique line.
2. Given a line and a point, there is a unique line through the point that is parallel to the given line.
3. There exist three points that are not all contained in a line.
|
We will say in this case that the affine plane
- has order n, and
- has a projective extension.
One can wonder what happens to the non-existent parallelism of the original projective plane: That is, in the projective plane there are no parallel classes (that is, sets of parallel lines) since no line has a line parallel to it. But we have removed n + 1 lines, and so now there may be some parallel classes, and indeed there are. We can see that in an affine plane of order n,
- each line is incident with n many points,
- each point is incident with n + 1many lines,
- there are n2 many points,
- there are n2 + n many lines,
- each line has n - 1 many lines parallel to it,
- and there are n + 1 parallel classes.
These observations might lead one to make up a geometry where there can be "tons" of parallel lines. More precisely, in the euclidean plane that is familiar to us from high school geometry (where there are, of course, infinitely many points) we know that given a line and a point not on that line, there is one and only one line through the point that is parallel to our original line. What we are asking for then is a geometry where there can be more than one line through this point that does not intersect our original line!
We will begin our journey much as we began our journey into projective geometry, by first looking at finite geometries. Let us propose the following axioms for our new geometry, geometries that we will call hyperbolic planes:
HYPERBOLIC PLANES
1. Two distinct points are contained in a unique line.
2. Associated to any non-incident point-line pair (p, l) is a set consisting of two or more lines, each line of which is incident with p but not l.
3. If a subspace X (X will be a subspace if p and q in X implies that the line pq is also in X) contains a triangle, then X must be the entire set of points.
|
What could finite hyperbolic planes look like? Some, but not all, t - (v, j, k) designs are hyperbolic planes.
t - (v, j, k) DESIGNS
1. Each line contains at least two points.
2. Any point is contained in any least two lines.
3. The total number of points is v.
4. Every line contains j-many points.
5. Given any t distinct points there are exactly k lines containing them.
|
A rather famous class of t - (v, j, k) designs are called unitals:
UNITALS
1. Unitals are 2 - (n3 + 1, n + 1, 1) designs where n > 1.
|
|
PROBLEM 16: (a) Which t - (v, j, k) designs are projective planes? Solution
16: (b) Which t - (v, j, k) designs are affine planes of order n? Solution
16: (c) Find the unique unital for n = 2, that is, find the unique (you do not have to show that it is unique) 2 - (9, 3, 1) design. Solution
16: (d) Show that all unitals with n > 2 are hyperbolic planes. Solution
Hint for 16(d): Use the following lemma.
Lemma. In a near-linear space, for any fixed point pi,
 .
|
|
References: Batten, Dembowski, Polster
|