Index
2. (a) The simplest example of a projective plane is the Fano plane:
Axiom 1 is easily satisfied, as a quick look at the figure will show that any two distinct points are connected by a unique line. By Axiom 2, two distinct lines must intersect at a unique point. In the Fano plane, there are 7 lines. It is easy to see that each intersects every other in a unique point. Finally, we have axiom 3: there exist 4 points of which no three are incident with the same line. We can easily show that this is satisfied because in the Fano plane, every line is incident with only 3 points. Thus, for any four points, no three are incident with the same line.
In additon, we will define the order of a projective plane to be some integer n, such that each line is incident with n+1 points and each point is incident with n+1 lines. In addition, the point set and the line set both have 
elements. Thus, the Fano plane has order 2.
2. (b) An affine plane is derived from an a projective plane by removing a line from the projective plane. Any two affine planes derived from a similar projective plane are isomorphic. Here are two examples of figures in the affine space created by removing a line from the Fano plane. These affine planes have order 2 because they are derived from a projective plane of order 2.
2. (c) Below is an example of a linear space:
2. (d) Below is a simple example of a design.
2. (e) Here is an example of an abstract plane configuration. This is actually a plane configuration (we can omit the word abstract) because this figure can be represented on the Euclidean plane using only straight lines. Note that any projective plane is an abstract plane configuation (not necessarily a plane configuration). In the case pictured, you will see that p = 3, l = 3, n = 2, m = 2.
2. (f) Below is an example of a generalized quadrangle.
2. (g) Below is an example of a biplane.
Problems -
(1) Read sections 1.2 and 1.3 from Polster, paying close attention to all definitions given. Then
- (a) give an example of a nonhomogeneous geometry (along with the axioms for that geometry) plus the line pencils for your example,
- (b) and give an example of a homogeneous geometry orther than a projective or affine plane (along with the axioms for that geometry) plus the line pencils for your example.
(2) Read section 3.1.1 from Dorwart/Berger on designs.
(3) Read sections 3.1 and 3.3 from Polster (many examples of designs are given).
(4) Read section 15.1 from Polster, and then visit the web site of the Set Game Company, and download SET Lite® (click on "downloads") and play a few games.
|
References: Batten, Polster.